text:
\section{Introduction} \label{sec:1}
As of today, it is not known whether there are
infinitely many prime twins $(p,\,p+2)$, or prime pairs
$(p,\,p+2r)$ with given $r>0$. However, using a
hypothesis on (weighted) equidistribution of primes in
arithmetic progressions, Goldston, Pintz and Yildirim
\cite{GPY05} have recently shown that there are infinitely
many pairs of primes differing by at most sixteen; see also
Goldston, Motohashi, Pintz and Yildirim \cite{GM06} and the
exposition by Soundararajan \cite{So07}. Let
$$\pi_{2r}(x) =\{\#\,\mbox{prime
pairs}\;(p,\,p+2r)\;\mbox{with}\; p\le x\}.$$
Around 1920 Viggo Brun used what is now called Brun's
sieve to prove that $\pi_2(x)=\cal{O}(x/\log^2 x)$.
In 1923 Hardy and Littlewood published a
long paper \cite{HL23} on the Goldbach problems
and on prime pairs, prime triplets, etc. For prime pairs
they conjectured the asymptotic formula
\begin{equation} \label{eq:1.1}
\pi_{2r}(x)\sim 2C_{2r}{\rm
li}_2(x)=2C_{2r}\int_2^x\frac{dt}{\log^2 t}\sim
2C_{2r}\frac{x}{\log^2 x}
\end{equation}
as $x\to \infty$. Here
\begin{equation} \label{eq:1.2}
C_2 = \prod_{p\,{\rm
prime},\,p>2}\,\left\{1-\frac{1}{(p-1)^2}\right\}
\approx 0.6601618,
\end{equation}
and
\begin{equation} \label{eq:1.3}
C_{2r} = C_2\prod_{p|r,\,p>2}\,\frac{p-1}{p-2}.
\end{equation}
Thus, for example, $C_4=C_8=C_2$, $C_6=2C_2$,
$C_{10}=(4/3)C_2$. There is a great deal of numerical
support for the prime-pair conjecture (PPC). On the Internet
one finds counts  of prime twins for $p$ up to $5\cdot
10^{15}$ by T.\ R.\ Nicely \cite{Ni05}. In Amsterdam Fokko
van de Bult \cite{Bu07} has recently counted the prime pairs
$(p,\,p+2r)$ with $2r\le 10^3$ and $p\le
x=10^3,\,10^4,\,\cdots,\,10^8$. Table $1$ is based on
his work. The bottom line shows (rounded) values $L_2(x)$
of the comparison function $2C_2{\rm li}_2(x)$. The table
supports the conjecture that for every $r$ and $\varepsilon>0$
\begin{equation} \label{eq:1.4}
\pi_{2r}(x)-2C_{2r}{\rm li}_2(x)\ll x^{(1/2)+\varepsilon}.
\end{equation}
Here the symbol $\ll$ is shorthand for the
$\cal{O}$-notation.

\begin{table} \label{table:1}
\begin{tabular}{llllllll}
$2r\backslash x$ & $10^3$ & $10^4$ & $10^5$ & $10^6$ & $10^7$
& $10^8$ & $C_{2r}/C_2$ \\
      &     &       &        &        &        &         &     \\
2     & 35  & 205   & 1224   & 8169   & 58980  & 440312  & 1   \\
4     & 41  & 203   & 1216   & 8144   & 58622  & 440258  & 1   \\
6     & 74  & 411   & 2447   & 16386  & 117207 & 879908  & 2   \\
8     & 38  & 208   & 1260   & 8242   & 59595  & 439908  & 1   \\
10    & 51  & 270   & 1624   & 10934  & 78211  & 586811  & 4/3 \\
12    & 70  & 404   & 2421   & 16378  & 117486 & 880196  & 2   \\
14    & 48  & 245   & 1488   & 9878   & 70463  & 528095  & 6/5 \\
16    & 39  & 200   & 1233   & 8210   & 58606  & 441055  & 1   \\
18    & 74  & 417   & 2477   & 16451  & 117463 & 880444  & 2   \\
20    & 48  & 269   & 1645   & 10972  & 78218  & 586267  & 4/3 \\
22    & 41  & 226   & 1351   & 9171   & 65320  & 489085  & 10/9\\
24    & 79  & 404   & 2475   & 16343  & 117342 & 880927  & 2   \\
30    & 99  & 536   & 3329   & 21990  & 156517 & 1173934 & 8/3 \\
210   & 107 & 641   & 3928   & 26178  & 187731 & 1409150 & 16/5\\
      &     &       &        &        &        &         &     \\
$L_2(x)\,$: & 46 & 214 & 1249 & 8248   & 58754  & 440368 &     \\
       &     &       &        &        &        &        &
\end{tabular}
\caption{Counting prime pairs}
\end{table}

Sieve methods have become an important part of prime-number
theory. Using an advanced sieve, Jie Wu \cite{Wu04} has
shown that $\pi_2(x)<6.8\,C_2\,x/\log^2x$ for all
sufficiently large $x$. The best result in  the other
direction is  J.\ R.\ Chen's \cite{Che73}: if $N(x)$ denotes the
number of primes $p\le x$ for which $p+2$ has at most two
prime factors, then $N(x)\ge cx/\log^2 x$ for some $c>0$.
There are related results for prime pairs $(p,\,p+2r)$. In
particular, for every $\varepsilon>0$ there is an $x_0=x_0(\varepsilon)$
independent of $r$ such that
\begin{equation} \label{eq:1.5}
\pi_{2r}(x)\le(8+\varepsilon)C_{2r}\,x/\log^2x\quad\mbox{for
all}\;\;x\ge x_0;
\end{equation}
see the book Sieve Methods by Halberstam and Richert
\cite{HR74}. We will also use the fact that the prime-pair
constants $C_{2r}$ have {\it mean value one}, for which
Tenenbaum \cite{Te06} has proposed an elegant proof. There is
a strong estimate in the work of Bombieri and Davenport
\cite{BD66}, which was sharpened by Friedlander and Goldston
\cite{FG95} to
\begin{equation} \label{eq:1.6}
S_m=\sum_{r=1}^m C_{2r}=m-(1/2)\log m+\cal{O}\{\log^{2/3}
(m+1)\}.
\end{equation}

Starting with Montgomery's work \cite{Mo73} one has
realized that there is a deep connection between the
prime-pair conjectures and the fine distribution of the
complex zeros of the zeta function. Goldston in California
has been an important contributor to the subject, cf.\
\cite{GM87}, \cite{GGOS00}; several papers exploit the PPC
to obtain plausible results on zeta's zeros.
Following a lead of Arenstorf \cite{Ar04} we will use a
Wiener--Ikehara theorem to study prime pairs; the two-way
form below is due to the author \cite{Ko05}.
\begin{theorem} \label{the:1.1}
Let $\sum_{n=1}^\infty a_n/n^{w}$ with $a_n\ge 0$ converge
to a sum function $f(w)$ for $w=u+iv$ with $u>1$. Then
\begin{equation} \label{eq:1.7}
\sum_{n\le x}\,a_n\,\sim\, Ax\quad\mbox{as}\;\;x\to\infty
\end{equation}
if and only if for $u\searrow 1$, the difference
\begin{equation} \label{eq:1.8}
f(u+iv)-\frac{A}{u+iv}=g(u+iv)
\end{equation}
has a distributional limit $g(1+iv)$, which on every finite
interval $(-B,B)$ coincides with a pseudofunction (that
may {\rm a priori} depend on $B$).
\end{theorem}
Ikehara \cite{Ik31} and Wiener \cite{Wi32} obtained
(\ref{eq:1.7}) under the hypothesis that $g(w)$ has an
analytic or continuous extension to the half-plane $\{u\ge
1\}$. The condition $\sum_{n\le x}\,a_n=\cal{O}(x)$ would
ensure that $f(u+iv)$ and $g(u+iv)$ have a distributional
limit as $u\searrow 1$. A pseudofunction is the
distributional Fourier transform of a bounded function
which tends to zero at $\pm\infty$; locally, such a
distribution is given by trigonometric series with
coefficients that tend to zero. A pseudofunction cannot
have pole-type singularities. In the case $a_n\ge 0$, local
pseudofunction boundary behavior of $g(w)$ in (\ref{eq:1.8})
implies that
\begin{equation} \label{eq:1.9}
(w-w_0)g(w)\to 0
\end{equation}
for angular approach of $w$ (from the right) to any point
$w_0$ on the line $\{u=1\}$; cf.\ \cite{Ko02},
or \cite{Ko04}, Theorem III.3.1.

\setcounter{equation}{0}
\section{Basic auxiliary functions}
\label{sec:2}
For analytic formulation of the general PPC one may
introduce the sums
$$\theta_{2r}(x)=\sum_{p,\,p+2r\,{\rm prime};\;p\le
x}\,\log^2 p.$$
Relation (\ref{eq:1.1}) is equivalent to the asymptotic
formula
$$\theta_{2r}(x)\sim 2C_{2r}x\quad\mbox{as}\;\;x\to\infty.$$
By the Wiener--Ikehara theorem this relation holds
if and only if the function
$$\tilde D_{2r}(w)=\sum_{p,\,p+2r\,{\rm prime}}\,\frac{\log^2
p}{p^w}$$
can be written as $2C_{2r}/(w-1)+g_{2r}(w)$, where
$g_{2r}(w)$ has `good boundary behavior' as $u\searrow 1$.

At this stage it is convenient to replace
$\theta_{2r}(x)$ and $\tilde D_{2r}(w)$ by functions with
similar behavior that involve von Mangoldt's function
$\Lambda(n)$. We recall its generating Dirichlet series; using
the Euler product for $\zeta(w)$,
$$\sum_{n=1}^\infty\,\frac{\Lambda(n)}{n^w}=-\frac{\zeta'(w)}{\zeta(w)}
=\sum_{p\,{\rm prime}}\,(\log p)
\bigg(\frac{1}{p^w}+\frac{1}{p^{2w}}+\cdots\bigg).$$
One has $\Lambda(k)=\log p$ if
$k=p^\alpha$ with $p$ prime, and $\Lambda(k)=0$ if $k$ is not a
prime power. Since there are only $\cal{O}(\sqrt x)$ prime
powers $p^\alpha\le x$ with $\alpha\ge 2$, the difference between
\begin{equation} \label{eq:2.1}
\psi_{2r}(x)\stackrel{\mathrm{def}}{=}
\sum_{n\le x}\,\Lambda(n)\Lambda(n+2r)
\end{equation}
and $\theta_{2r}(x)$ is not much larger than $\sqrt x$.
Thus the PPC is also equivalent to the relation
\begin{equation} \label{eq:2.2}
\psi_{2r}(x)\sim 2C_{2r}x\quad\mbox{as}\;\;x\to\infty.
\end{equation}
Similarly, the function
\begin{equation} \label{eq:2.3}
D_{2r}(s)\stackrel{\mathrm{def}}{=}\sum_{n=1}^\infty
\,\frac{\Lambda(n)\Lambda(n+2r)}{n^s(n+2r)^s}\qquad(s=\sigma+i\tau,\,
\sigma>1/2)
\end{equation}
behaves in the same way as $\tilde D_{2r}(2s)$ when $2\sigma$
is close to $1$. Setting
\begin{equation} \label{eq:2.4}
D_{2r}(s)-\frac{C_{2r}}{s-1/2}=G_{2r}(s),
\end{equation}
the Wiener--Ikehara theorem with $2s$ instead of $w$
shows that the PPC (\ref{eq:2.2}) is equivalent to good
boundary behavior of $G_{2r}(s)$ as
$\sigma\searrow 1/2$.

\medskip\noindent {\scshape Combinations}.
In order to profit from the fact that the constants $C_{2r}$
have mean value $1$ it helps to study sums
$\sum_{2r\le\lambda}\,D_{2r}(s)$ for large values of $\lambda$.
Indeed, under the PPC, their boundary behavior should be
roughly like that of $(\lambda/2)/(s-1/2)$. In this spirit we
will study manageable combinations $V^\lambda(s)$ of functions
$D_{2r}(s)$ with nonnegative coefficients. They are derived
from a certain repeated complex integral $T^\lambda(s)$ (see
Section \ref{sec:5}) which extends and modifies an integral
of Arenstorf \cite{Ar04}. It involves a parameter
$\lambda>0$ and a parameter function $E^\lambda$; the
resulting formula for $V^\lambda(s)$ is
\begin{equation} \label{eq:2.5}
V^\lambda(s) \stackrel{\mathrm{def}}{=} 2\sum_{0<2r\le\lambda}
\,E^\lambda(2r)D_{2r}(s) = T^\lambda(s) - D_0(s) + H^\lambda(s).
\end{equation}
Here the function $D_{2r}(s)$ is given by (\ref{eq:2.3}),
also when $r=0$, and $H^\lambda(s)$ is holomorphic for
$\sigma>0$. The parametric function
$E^\lambda(\nu)=E(\nu/\lambda)$ acts as a sieving device. The basic
function $E(\nu)$ is taken even, with compact support,
Lipschitz continuous and decreasing on $[0,\infty)$. For
convenience $E(\nu)$ is normalized so that its support is
$[-1,1]$ and $E(0)=1$. The simplest sieving function
$E^\lambda(\nu)$ is given by the Fourier transform of the
Fej\'{e}r kernel for ${\Bbb R}$,
$$E^\lambda_F(\nu) = \frac{1}{\pi}\int_0^\infty\frac{\sin^2(\lambda
t/2)}{\lambda(t/2)^2}\cos \nu t\,dt =
\left\{\begin{array}{ll} 1-|\nu|/\lambda & \mbox{for
$|\nu|\le\lambda$,}\\  0 & \mbox{for $|\nu|\ge\lambda$.}
\end{array}\right.$$
This function is adequate if one is willing to use Riemann's
Hypothesis (RH) in the proof of the main theorem; cf.\ the
manuscript \cite{Ko07}. In the present paper we will prove
the main result without appealing to RH, but for that have to
require that $E$ be sufficiently {\it smooth}. More
precisely, we suppose that $E$, $E'$ and $E''$ are absolutely
continuous with $E'''$ of bounded variation. One could for
example use the Fourier transform of the Jackson kernel
for ${\Bbb R}$,
\begin{align}
E^\lambda_J(\nu) &=
\frac{3}{4\pi}\int_0^\infty\frac{\sin^4(\lambda t/4)}
{\lambda^3(t/4)^4}\cos \nu t\,dt
\notag \\ &=\left\{\begin{array}{ll}
1-6(\nu/\lambda)^2+6(|\nu|/\lambda)^3 & \mbox{for
$|\nu|\le\lambda/2$},\\ 2(1-|\nu|/\lambda)^3 & \mbox{for
$\lambda/2\le|\nu|\le\lambda$},\\ 0 & \mbox{for $|\nu|\ge\lambda$.}
\end{array}\right.\notag
\end{align}

The PPC and the mean value $1$ of the constants $C_{2r}$ lead
one to expect that for large $\lambda$, $V^\lambda(s)$ has a
first-order pole at $s=1/2$ with residue
\begin{equation} \label{eq:2.6}
2\sum_{0<2r\le\lambda}\,E(2r/\lambda)C_{2r}\approx\lambda\int_0^1
E(\nu)d\nu \stackrel{\mathrm{def}}{=} A^E\lambda.
\end{equation}

For the following we need a Mellin transform associated
with the Fourier transform of the kernel $E^\lambda$:
\begin{equation} \label{eq:2.7}
M^\lambda(z)=M^\lambda_E(z) \stackrel{\mathrm{def}}{=}
\frac{1}{\pi}\int_0^\infty \hat E^\lambda(t)t^{-z}dt,\quad
-1<x={\rm Re}\,z<1.
\end{equation}
\begin{proposition} \label{prop:2.1}
For our smooth $E$, the Mellin transform has a meromorphic
extension to the half-plane $\{x>-3\}$, given by
\begin{equation} \label{eq:2.8}
M^\lambda(z) = \frac{2}{\pi}\lambda^z\Gamma(-z-3)\sin(\pi z/2)
\int_0^{1+} \nu^{z+3}dE'''(\nu)=\lambda^zM(z),
\end{equation}
say. It has poles (of the first order) at $z=1,\,3,\,\cdots$.
The residue at $z=1$ is $-(2\lambda/\pi)A^E=-(2\lambda/\pi)\int_0^1
E(\nu)d\nu$ and $M^\lambda(0)=1$. For fixed $\lambda$ and any
constant $C$ one has the majorization
\begin{equation} \label{eq:2.9}
M^\lambda(x+iy)\ll(|y|+1)^{-x-7/2}\quad\mbox{for}\;\;-3<x\le
C,\;\; |y|\ge 1.
\end{equation}
\end{proposition}
\begin{proof}
The Fourier transform $\hat E^\lambda(t)$ is
$\cal{O}\{(|t|+1)^{-2}\}$. By (\ref{eq:2.7}), initially
taking $0<x<1$ and using the Mellin transform of $\cos\nu t$,
cf.\ Section \ref{sec:4},
\begin{align}
M^\lambda(z) &= \frac{1}{\pi}\int_0^\infty
t^{-z}dt\cdot 2\int_0^\lambda E^\lambda(\nu)(\cos t\nu)d\nu
\notag \\ & =\frac{2}{\pi}\int_0^\lambda E^\lambda(\nu)d\nu
\int_0^{\infty-}(\cos\nu t)t^{-z}dt\notag\\ &=
\frac{2}{\pi}\Gamma(1-z)\cos\{\pi(1-z)/2\}\int_0^\lambda
E^\lambda(\nu)\nu^{z-1}d\nu \notag \\ &=
\frac{2}{\pi}\lambda^z\Gamma(1-z)\sin(\pi z/2)\int_0^1
E(\nu)\nu^{z-1}d\nu.\notag
\end{align}
For $x>0$ and smooth $E$, the final integral may also be
written as
\begin{align}
-\frac{1}{z}\int_0^1 \nu^zdE(\nu) &=
\frac{1}{z(z+1)}\int_0^{1}\nu^{z+1}dE'(\nu) \notag \\ &=
\frac{1}{z(z+1)(z+2)(z+3)}\int_0^{1+}\nu^{z+3}dE'''(\nu).
\notag
\end{align}
This is enough to prove (\ref{eq:2.8}), hence $M^\lambda(z)$
has a  meromorphic extension to the half-plane $\{x>-3\}$.
The poles  of $\Gamma(-z-3)$ at $z=-2,\,0,\,2,\,\cdots$ are
cancelled by zeros of $\sin(\pi z/2)$ and the pole at $z=-1$
is cancelled by the zero of $\int_0^1\nu^{z+1}dE'(\nu)$ at
that point. The formulas also show that
$M^\lambda(0)=1$ and that the residue at the pole $z=1$ is equal
to $-(2\lambda/\pi)\int_0^1 E(\nu)d\nu$. The order estimate
(\ref{eq:2.9}) follows from the standard inequalities
\begin{equation} \label{eq:2.10}
\Gamma(z)\ll |y|^{x-1/2}e^{-\pi|y|/2},\quad\sin(\pi z/2)\ll
e^{\pi|y|/2}
\end{equation}
which are valid for $|x|\le C$ and $|y|\ge 1$. The inequality
for $\Gamma(z)$ follows from Stirling's formula for complex
$z$; see formula (\ref{eq:8.3}) below.
\end{proof}

\setcounter{equation}{0}
\section{Results}\label{sec:3}
Our results involve the complex zeros $\rho$ of the
zeta function. Taking multiplicities into account, the
zeros above the real axis will be arranged according to
non-decreasing imaginary part:
$$\rho=\rho_n=\beta_n+i\gamma_n,\;\;0<\gamma_1\approx
14<\gamma_2\approx 21\le\cdots,\;\;n=1,\,2,\,\cdots$$
(with $\beta_n=1/2$ as far as zeros have been computed); we
write $\overline\rho_n=\rho_{-n}$. The theorem below involves
the sum
\begin{align} & \label{eq:3.1}
\Sigma^\lambda(s) \stackrel{\mathrm{def}}{=}
\left\{\frac{\zeta'(s)}{\zeta(s)}\right\}^2 \notag \\ & \quad
+2\,\frac{\zeta'(s)}{\zeta(s)}\,
\sum_{\rho}\,\Gamma(\rho-s)M^\lambda(\rho-s)
\cos\{\pi(\rho-s)/2\} \\ &
+ \sum_{\rho,\,\rho'}\,\Gamma(\rho-s)\Gamma(\rho'-s)
M^\lambda(\rho+\rho'-2s)\cos\{\pi(\rho-\rho')/2\},\notag
\end{align}
where $M^\lambda(\cdot)$ is given by (\ref{eq:2.7}). It is
convenient to denote the sum of the first two terms by
$\Sigma^\lambda_1(s)$ and to set the double sum equal to
$\Sigma^\lambda_2(s)$. Results from Section \ref{sec:2} show that
$\Sigma^\lambda_1(s)$ defines a meromorphic function for $\sigma<3$
whose only poles in the strip $\{0<\sigma<1\}$ occur at the
complex zeros of $\zeta(\cdot)$. Under RH the double
series, in which $\rho$ and $\rho'$ both run over zeta's
complex zeros, is absolutely convergent for $1/2<\sigma<1$;
cf.\ Lemma \ref{lem:4.2} below. Without RH the double sum may
be interpreted as a limit of sums over the zeros $\rho$,
$\rho'$ with $|{\rm Im}\,\rho|$,
$|{\rm Im}\,\rho'|<R$; it will follow from Theorem
\ref{the:3.1} that the combination $\Sigma^\lambda(s)$ is in any
case holomorphic for $1/2<\sigma<1$.

The formula for $V^\lambda(s)$ in (\ref{eq:2.5}) contains the
function $T^\lambda(s)$ for which a repeated complex integral is
introduced in Section \ref{sec:5}. Moving the paths of
integration in this integral and using the residue theorem
one obtains
\begin{theorem} \label{the:3.1}
For any $\lambda>0$, any smooth sieving function $E$, and for
$s=\sigma+i\tau$ with $1/2<\sigma<1$ there are holomorphic
representations
\begin{align}  \label{eq:3.2}
V^\lambda(s) &= 2\sum_{0<2r\le\lambda}\,E(2r/\lambda)D_{2r}(s) \notag \\
&= \frac{-1/4}{(s-1/2)^2}+\frac{A^E\lambda}{s-1/2}
+\Sigma^\lambda(s)+H^\lambda(s) \\ &=
\frac{A^E(\lambda-1)}{s-1/2}+\Sigma^\lambda(s)-\Sigma^1(s)+H^\lambda(s),
\notag \label{eq:3.2}
\end{align}
where $A^E=\int_0^1 E(\nu)d\nu$ and $\Sigma^\lambda(s)$ is given by
$(\ref{eq:3.1})$ (with proper interpretation of the double
sum); the various functions $H^\lambda(s)$ are analytic for
$1/2\le\sigma<1$, and for $1/4<\sigma<1$ under RH. On the interval
$\{1/2\le s\le 3/4\}$ one has $H^\lambda(s)=\cal{O}(\lambda\log\lambda)$
as $\lambda\to \infty$.
\end{theorem}
The (extended) Wiener--Ikehara Theorem will
now show that the Hardy--Littlewood conjectures for prime
pairs $(p,\,p+2r)$ are true if and only if the differences
$\Sigma^\lambda(s)-\Sigma^1(s)$ exhibit certain specific boundary
behavior  as $\sigma\searrow 1/2$; cf.\ (\ref{eq:2.4}). To make
this precise, define
\begin{equation} \label{eq:3.3}
R(\lambda)=2\sum_{0<2r\le\lambda}
\,E(2r/\lambda)C_{2r}-A^E(\lambda-1).
\end{equation}
By induction Theorems \ref{the:3.1} and \ref{the:1.1} imply
\begin{corollary} \label{cor:3.2}
Suppose that the prime-pair conjecture for pairs
$(p,\,p+2r)$ is true for every $r<m$. Then the PPC for
prime pairs $(p,\,p+2m)$ is true if and only if for some
(or every) smooth function $E$ and some (or every)
number $\lambda\in(2m,2m+2]$, the function
\begin{equation} \label{eq:3.4}
G^\lambda(s) \stackrel{\mathrm{def}}{=}
\Sigma^\lambda(s)-\Sigma^1(s)-\frac{R(\lambda)}{s-1/2}
\end{equation}
has good (local pseudofunction) boundary behavior as
$\sigma\searrow 1/2$.
\end{corollary}
The double series in (\ref{eq:3.1}) defines $\Sigma^\lambda_2(s)$
as a meromorphic function for $1/2<\sigma<1$ whose poles occur
at complex zeros of $\zeta(\cdot)$, and these poles are
cancelled by those of $\Sigma^\lambda_1(s)$. Formally there is
cancellation also at the other complex zeros of
$\zeta(\cdot)$. Turning to the function
$G^\lambda(s)$, note that by (\ref{eq:3.2}), it {\it does}
have good boundary behavior when $\lambda\le 2$; under RH, it
will even be analytic for $1/4<\sigma<1$. Indeed,
$V^\lambda(s)=0$ for $\lambda\le 2$. These observations support the
following
\begin{conjecture} \label{con:3.3}
For every $\lambda>0$ and every $E$, the function $G^\lambda(s)$ in
$(\ref{eq:3.4})$ has an analytic continuation to the strip
$\{1/4<\sigma<1\}$.
\end{conjecture}
\noindent If this is true, the counting functions
$\pi_{2r}(x)$ all satisfy estimates of type (\ref{eq:1.4}).

\smallskip\noindent{\scshape Conditional abundance of prime
pairs.} It will follow from Section \ref{sec:7} that the part
of the sum $\Sigma^\lambda_2(s)$ in which ${\rm Im}\,\rho$ and
${\rm Im}\,\rho'$ have the same sign defines a meromorphic
function for $1/2\le\sigma<1$ whose only poles occur at
complex zeros of $\zeta(\cdot)$. Thus for a study of its
pole-type behavior near the point
$s=1/2$, the double sum $\Sigma^\lambda_2(s)$ in (\ref{eq:3.1})
may be reduced to the sum $\Sigma^\lambda_3(s)$ in which ${\rm
Im}\,\rho$ and ${\rm Im}\,\rho'$ have opposite sign. Hence
in the study of the PPC under RH, the {\it differences} of
zeta's zeros on the same side of the real axis play a key
role.
\begin{theorem} \label{the:3.4}
Assume RH. Then the pole-type behavior of $\Sigma^\lambda_3(s)$ and
$\Sigma^\lambda(s)$ as $s\searrow 1/2$ is the same as that of the
reduced sum
\begin{equation} \label{eq:3.5}
\Sigma^\lambda_4(s) = 2\pi\sum_{\gamma,\,\gamma';\,|\gamma'-\gamma|<\gamma^{1/2}}\,
\gamma^{-2s+i(\gamma-\gamma')}M^\lambda\{1-2s+i(\gamma-\gamma')\},
\end{equation}
where $\gamma$ and $\gamma'$ run over the imaginary parts of
the zeros of $\zeta(\cdot)$ in the upper half-plane.
\end{theorem}
The expression in (\ref{eq:3.5}) is reminiscent of the
pair-correlation function of zeta's complex zeros which
was studied by Montgomery \cite{Mo73} et al. Since the
constants $C_{2r}$ have mean value $1$, the function $R(\lambda)$
in (\ref{eq:3.3}) is $o(\lambda)$ as $\lambda\to\infty$; cf.\
(\ref{eq:2.6}). [By (\ref{eq:1.6}) it will even be
$\cal{O}(\log\lambda)$.] The corresponding hypothesis below
regarding $\Sigma^\lambda(s)-\Sigma^1(s)$ would follow from a plausible
extension of Montgomery's work; see Section \ref{sec:9}.
\begin{hypothesis} \label{hyp:3.5}
For smooth $E$ the `upper residue'
\begin{equation} \label{eq:3.6}
\omega(\lambda)=\omega^E(\lambda)=\limsup_{s\searrow
1/2}\,(s-1/2)\{\Sigma^\lambda(s)-\Sigma^1(s)\}
\end{equation}
is $o(\lambda)$ as $\lambda\to\infty$.
\end{hypothesis}
It follows from (\ref{eq:3.2}) and (\ref{eq:1.5}) that
$\omega(\lambda)=\cal{O}(\lambda)$. If Hypothesis \ref{hyp:3.5} is true
there will be an abundance of prime pairs:
\begin{theorem} \label{the:3.6}
Assume Hypothesis \ref{hyp:3.5}. Then for
every $\varepsilon>0$, there is a positive integer $m$,
depending on $\omega(\cdot)$ and $\varepsilon$, such that
\begin{equation} \label{eq:3.7}
\limsup_{x\to \infty}\,\frac{1}{m}
\sum_{r\le m}\,\frac{\pi_{2r}(x)}{x/\log^2x}>2-\varepsilon.
\end{equation}
\end{theorem}
\noindent Here the constant $2$ would be optimal.

We finally mention an interesting positivity property of
certain double sums $\Sigma^\lambda_2(s)$ in (\ref{eq:3.1}):
\begin{proposition} \label{prop:3.7}
Let $E^\lambda$ be a sieving function (such as
$E^\lambda_J$) for which $\hat E^\lambda(t)\ge 0$. Then
$\Sigma^\lambda_2(s)\ge 0$ when $1/2<s<1$.
\end{proposition}
This positivity  and a speculative equidistribution result
for prime pairs with different values of $2r$
would also imply that there is an abundance of prime pairs;
see Section
\ref{sec:10}.

\setcounter{equation}{0}
\section{Complex representation for $E^\lambda(\alpha-\beta)$}
\label{sec:4}
For the discussion of $T^\lambda(s)$ in Section \ref{sec:5} we
need a complex integral for the sieving function
$E^\lambda(\alpha-\beta)$ in which $\alpha$ and $\beta$ occur separately.
It is obtained from the representation of $E^\lambda(\alpha-\beta)$ as
an inverse Fourier (cosine) transform:
$$E^\lambda(\alpha-\beta)=\frac{1}{\pi}
\int_0^\infty \hat E^\lambda(t)\cos\{(\alpha-\beta)t\}dt$$
and a repeated complex integral for
$$\cos\{(\alpha-\beta)t\}=\cos\alpha t\cos\beta t +\sin\alpha t\sin\beta
t.$$
To set the stage we start with a complex representation
for $\cos\alpha$ and $\sin\alpha$ with $\alpha>0$. Setting $z=x+iy$
we write $L(c)$ for the `vertical line' $\{x=c\}$; the
factor $1/(2\pi i)$ in complex integrals will be omitted.
Thus
$$\int_{L(c)}f(z)dz\stackrel{\mathrm{def}}{=}
\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}f(z)dz.$$
Mellin inversion of the improper Euler integral
$$\int_0^{\infty-}(\cos\alpha)\alpha^{z-1}d\alpha=\Gamma(z)\cos(\pi
z/2)\qquad(0<x<1)$$
now gives the improper complex integral
\begin{align}
\cos\alpha &= \int_{L(c)}^*\Gamma(z)\alpha^{-z}\cos(\pi
z/2)dz \notag \\ &= \lim_{A\to\infty}\frac{1}{2\pi
i}\int_{c-iA}^{c+iA}\,\cdots\quad(0<c<1/2);\notag
\end{align}
there is a similar representation for $\sin\alpha$.
It is important for us to have absolutely convergent
integrals. We therefore replace the line $L(c)$ by a
path $L(c,B)=L(c_1,c_2,B)$ with suitable $c_1<c_2$ and
$B>0$ (cf.\ Figure \ref{fig:1}):
\begin{equation} \label{eq:4.1}
L(c,B)=\left\{\begin{array}{lllll}
\mbox{$\quad$the half-line} & \mbox{$\{x=c_1,\,-\infty<y\le
-B\}$}\\
\mbox{$+\;$the segment} & \mbox{$\{c_1\le x\le
c_2,\,y=-B\}$}\\
\mbox{$+\;$the segment} & \mbox{$\{x=c_2,\,-B\le y\le B\}$}\\
\mbox{$+\;$the segment} & \mbox{$\{c_2\ge x\ge
c_1,\,y=B\}$}\\
\mbox{$+\;$the half-line} & \mbox{$\{x=c_1,\,B\le
y<\infty\}$.}
\end{array}\right.
\end{equation}
Taking $c_1<-1/2$, $c_2>0$ and using formula (\ref{eq:2.10}),
one thus obtains the absolutely convergent repeated integral
\begin{align}
\cos\{(\alpha-\beta)t\} &= \int_{L(c,B)}
\Gamma(z)\alpha^{-z}t^{-z}dz\,\cdot \notag \\&  \quad\cdot
\int_{L(c,B)}\Gamma(w)\beta^{-w}t^{-w}\cos\{\pi(z-w)/2\}dw.
\notag
\end{align}
Multiplying both sides by $\hat E^\lambda(t)$,
integrating over $\{0<t<\infty\}$, inverting order of
integration and using formula (\ref{eq:2.7}) for
$M^\lambda(\cdot)$, one obtains
\begin{proposition} \label{prop:4.1}
Let $\alpha,\,\beta>0$, $-3/2<c_1<0<c_2<1/2$ and $B>0$. Then
for smooth sieving functions $E^\lambda(\cdot)$ as in Section
\ref{sec:2} one has the representation
\begin{align} \label{eq:4.2}
E^\lambda(\alpha-\beta) & = \int_{L(c,B)}\Gamma(z)\alpha^{-z}dz
\int_{L(c,B)}\Gamma(w)\beta^{-w}\,\cdot\notag
\\ & \quad\cdot M^\lambda(z+w)\cos\{\pi(z-w)/2\}\,dw.
\end{align}
\end{proposition}
\begin{figure}[htb]
$$\includegraphics[width=4cm]{Kor1.eps}
$$
\caption{The path $L(c_1,c_2,B)$ \label{fig:1}}
\end{figure}
To justify the operations and verify the absolute
convergence of the integral in (\ref{eq:4.2}) one may use
the estimate (\ref{eq:2.9}) and a simple lemma:
\begin{lemma} \label{lem:4.2}
For real constants $a,\,b,\,c$, the function
$$\phi(y,v)=(|y|+1)^{-a}(|v|+1)^{-b}(|y+v|+1)^{-c}$$
is integrable over ${\Bbb R}^2$ if (and only if) $a+b>1$,
$a+c>1$, $b+c>1$ and $a+b+c>2$. For integrability over
${\Bbb R}_+^2$ the condition $a+b>1$ may be dropped.
\end{lemma}
For verification let $A>0$. On the subset of ${\Bbb R}^2$
where $|y|\le A$, the condition $b+c>1$ is necessary and
sufficient for a finite $v$-integral. Similarly for $|v|\le
A$ and the condition $a+c>1$. When $|y+v|\le A$ one needs
the condition $a+b>1$ in the case of ${\Bbb R}^2$. For the
fourth condition one looks at the set where $v\ge y\ge 1$.
Setting $v=yr$ with new variable $r$,
$$\phi(y,v)dydv\asymp y^{-a}(yr)^{-b}\{y(r+1)\}^{-c}ydydr,$$
and the right-hand side is integrable over the set
$\{1<y<\infty,\,1<r<\infty\}$ if and only if $b+c>1$
and $a+b+c>2$. Similarly when $y$ and $v$ have opposite
sign; one may of course assume that $|y+v|>1$ then.

We turn to Proposition \ref{prop:4.1}. Setting $z=x+iy$,
$w=u+iv$, Stirling's formula and (\ref{eq:2.9}) give the
following majorant for the integrand in (\ref{eq:4.2}) on
the remote parts of the paths $L(c,B)$:
$$(|y|+1)^{c_1-1/2}(|v|+1)^{c_1-1/2}(|y+v|+1)^{-2c_1-7/2}.$$
For integrability one thus needs $-3<c_1<0$. The more
stringent requirements in the proposition
serve to justify inversion of the order of integration in a
triple integral and to keep the paths within the strip
$\{-3<X<1\}$ where $M^\lambda(Z)$ is known to be regular.

\setcounter{equation}{0}
\section{The complex integral for $T^\lambda(s)$} \label{sec:5}
The function $T^\lambda(s)$ in (\ref{eq:2.5}) is
defined by the integral below for $\sigma>1+|c_1|$, while for
$s$ with smaller real part it is defined by analytic
continuation;
\begin{align}  \label{eq:5.1}
\quad T^\lambda(s) &=
\int_{L(c,B)}\Gamma(z)\frac{\zeta'(z+s)}{\zeta(z+s)}\,dz
\int_{L(c,B)}\Gamma(w)\frac{\zeta'(w+s)}{\zeta(w+s)}\,
\cdot \notag \\ & \qquad\cdot
M^\lambda(z+w)\cos\{\pi(z-w)/2\}dw.
\end{align}
\begin{theorem} \label{the:5.1}
Let $-3/2<c_1<0<c_2<1/2$. Then the integral
$(\ref{eq:5.1})$ defines $T^\lambda(s)$ as a holomorphic
function of $s=\sigma+i\tau$ for $\sigma>1-c_1$. Assuming RH, the
integral gives $T^\lambda(s)$ as a holomorphic function for
$\sigma>\max\{(1/2)-c_1,1-c_2\}$ and $|\tau|<B$.

The integral has an analytic continuation to the half-plane
$\{\sigma>1/2\}$ given by the expansion
\begin{equation} \label{eq:5.2}
T^\lambda(s)=\sum_{k,l}\Lambda(k)\Lambda(l)k^{-s}l^{-s}E^\lambda(k-l).
\end{equation}
\end{theorem}
\begin{proof}[Discussion]
For $z\in L(c,B)$ and $\sigma>1-c_1$, the sum
$z+s$ will stay away from the poles of $\zeta'/\zeta$. Under
RH the same holds when $\sigma>\max\{(1/2)-c_1,1-c_2\}$
and $|\tau|<B$. Indeed, in that case $x+\sigma>1/2$ and also
$z+s\ne 1$: if $x+\sigma=1$, then $z$ must lie on the part of
$L(c,B)$ where $|y|\ge B$, and then $y+\tau\ne 0$. Similarly
for $w=u+iv$. The absolute convergence of the repeated
integral in (\ref{eq:5.1}) can be proved in the same way as
that of (\ref{eq:4.2}). Indeed, the quotient $(\zeta'/\zeta)(Z)$
grows at most logarithmically in $Y$ for $X\ge 1$, and for
$X\ge (1/2)+\eta$ under RH; cf.\ Titchmarsh \cite{Ti86}. The
holomorphy of the integral for $T^\lambda(s)$ now follows from
locally uniform convergence in $s$.

For the second part we substitute the Dirichlet series for
$(\zeta'/\zeta)(\cdot)$ into (\ref{eq:5.1}),
initially taking $\sigma>1-c_1$. Integrating term by term and
applying Proposition \ref{prop:4.1} one obtains the
expansion (\ref{eq:5.2}). Because $E^\lambda(k-l)\ne 0$
only for finitely many values of $k-l$, the
series represents a holomorphic function for $\sigma>1/2$;
cf.\ the proof of Theorem \ref{the:5.2} below. The sum of
the series provides an (the) analytic continuation of the
integral to the half-plane $\{\sigma>1/2\}$.
\end{proof}
We will now derive (\ref{eq:2.5}).
\begin{theorem} \label{the:5.2}
For arbitrary $\lambda>0$ and $\sigma>1/2$,
\begin{equation} \label{eq:5.3}
T^\lambda(s)=D_0(s)+2\sum_{0<2r\le\lambda}\,E(2r/\lambda)D_{2r}(s)
+H^\lambda_1(s),
\end{equation}
where $H^\lambda_1(s)$ is holomorphic for $\sigma>0$. On the real
interval $\{1/2\le s\le 3/4\}$ one has
$H^\lambda_1(s)=\cal{O}(\lambda)$ as $\lambda\to \infty$.
\end{theorem}
\begin{proof}
Taking $k=l$ in (\ref{eq:5.2}) one obtains the term
$D_0(s)$ in (\ref{eq:5.3}). For $|k-l|=2r$ one
obtains a constant multiple of $D_{2r}(s)$. The coefficient
is different from $0$ only if $2r<\lambda$ and in fact equal
to $2E(2r/\lambda)$. If
$|k-l|=2r-1$ one can have $\Lambda(k)\Lambda(l)\ne 0$ only if either
$k$ or $l$ is of the form $2^\alpha$ for some $\alpha>0$. The
resulting functions, for which $2r-1$ must be $<\lambda$, are
holomorphic for $\sigma>0$. For real $s\searrow 1/2$ the sum
$H^\lambda_1(s)$ of their values will be $\cal{O}(\lambda\log\lambda)$
as $\lambda\to \infty$.
\end{proof}
It remains to determine the analytic character of $D_0(s)$:
\begin{lemma} \label{lem:5.3}
One has
\begin{equation} \label{eq:5.4}
D_0(s)=\sum_{k=1}^\infty\,\frac{\Lambda^2(k)}{k^{2s}}
=\frac{1/4}{(s-1/2)^2}+H_0(s),
\end{equation}
where $H_0(s)$ is analytic for $\sigma\ge 1/2$, and for
$\sigma>1/4$ under RH.
\end{lemma}
Indeed, for $x>1$
\begin{align} &
\sum_k\Lambda^2(k)k^{-z} = \sum_p(\log^2 p)p^{-z}+H_1(z)
= -\frac{d}{dz}\sum_p(\log p)p^{-z}+H_1(z) \notag \\ &=
-\frac{d}{dz}\sum_k\Lambda(k)k^{-z}+H_2(z) =
\frac{d}{dz}\frac{\zeta'(z)}{\zeta(z)}+H_2(z)=
\frac{1}{(z-1)^2}+H_3(z),\notag
\end{align}
where $H_1(z)$ and $H_2(z)$ define holomorphic functions for
$x>1/2$, while $H_3(z)$ is holomorphic for $x\ge 1$, and
for $x>1/2$ under RH. Finally take $z=2s$.

\setcounter{equation}{0}
\section{Transformation of the integral for $T^\lambda(s)$}
\label{sec:6}
Taking $c_1,\,c_2$ and $s$ as in the first part of
Theorem \ref{the:5.1} we will move the paths of integration
in the integral for $T^\lambda(s)$, but first change variables.
Replacing $z$ by $z'-s$ and $w$ by $w'-s$ (and subsequently
dropping the primes), one obtains
\begin{align} \label{eq:6.1}
T^\lambda(s) & = \int_{L(c',B')}\Gamma(z-s)
\frac{\zeta'(z)}{\zeta(z)}\,dz
\int_{L(c',B')}\Gamma(w-s)\frac{\zeta'(w)}{\zeta(w)}\,
\cdot \notag \\ & \quad\; \cdot
\,M^\lambda(z+w-2s)\cos\{\pi(z-w)/2\}dw.
\end{align}
Here the paths of integration will initially depend on $s$:
$c'_1=c_1+s$, $c'_2=c_2+s$, and the horizontal segments
may be at different distances from the real axis. However,
by our standard estimates and Cauchy's theorem, one may
fix $c'_1=1$ and $c'_2=3/2$, say, use a constant $B'$
and take $1<\sigma<3/2$, $|\tau|<B'$. Observe that henceforth,
the point $s$ will be to the left of the paths.

Starting with (\ref{eq:6.1}), where we rename
$c'_1=1=c_1$, $c'_2=3/2=c_2$ and $B'=B$, the paths of
integration $L(c,B)$ will be moved across the poles
$s$, $1$ and $\rho$ to the line $L(0)$, the imaginary axis.
We first describe what happens when we move the
$w$-path:
\begin{equation} \label{eq:6.2}
T^\lambda(s) = \int_{L(c,B)}\cdots\,dz\int_{L(0)}\cdots\,dw
+U^\lambda_2(s)=U^\lambda_1(s) +U^\lambda_2(s),
\end{equation}
say, where by the residue theorem
\begin{equation} \label{eq:6.3}
U^\lambda_2(s) = \int_{L(c,B)}
\Gamma(z-s)\frac{\zeta'(z)}{\zeta(z)}\,J(z,s)dz,
\end{equation}
with
\begin{align} \label{eq:6.4}
J(z,s) &=
\frac{\zeta'(s)}{\zeta(s)}\,M^\lambda(z-s)\cos\{\pi(z-s)/2\}
\notag \\ &
\quad -\Gamma(1-s)M^\lambda(z+1-2s)\cos\{\pi(z-1)/2\} \\ &
\quad +\sum_\rho\,\Gamma(\rho-s)M^\lambda(z+\rho-2s)
\cos\{\pi(z-\rho)/2\}.\notag
\end{align}
By the usual estimates and Lemma \ref{lem:4.2} the
repeated integral $U^\lambda_1(s)$ defines a holomorphic
function for $1/2<\sigma<3/2$, hence by Theorem
\ref{the:5.1}, the function $U^\lambda_2(s)$ must have an
analytic continuation to the same strip!

For $1<\sigma<3/2$ the function $J(z,s)$ is holomorphic in
$z$ on and between the paths $L(c,B)$ and $L(0)$. If we
define $J(z,s)$ for $z\in L(c,B)$ by continuity at
$s=1$ and points $s=\rho$, it becomes holomorphic in $s$
for $3/4<\sigma<3/2$; apparent poles cancel each other.
What can we say about the integral for $U^\lambda_2(s)$? The
critical part is the one that
corresponds to the sum over $\rho=\beta\pm i\gamma$ in
(\ref{eq:6.4}). For $|y|>B\ge 2$ its integrand is majorized
by
$$\sum_\rho |y|^{c_1-\sigma-1/2}(\log|y|)|\rho|^{\beta-\sigma-1/2}
(|y+\rho|+1)^{-c_1-\beta+2\sigma-7/2}.$$
Since $c_1=1$, $0<\beta<1$ and $|\rho_n|\sim 2\pi n/\log n$ as
$n\to\infty$, the analog of Lemma \ref{lem:4.2} for the
integral of a sum proves the absolute convergence and
holomorphy of the integral when $3/4<\sigma<3/2$.

To justify the above application of the residue theorem
one may start with $w$-integrals over a sequence of
closed contours $W_R$, $B<R=R_k\to\infty$, whose upper part
is shown in Figure \ref{fig:2}.
\begin{figure}[htb]
$$\includegraphics[width=4cm]{Kor4.eps}
$$
\caption{Upper half of $W_R$ \label{fig:2}}
\end{figure}
Here the numbers $R=R_k$ are chosen `away from the numbers
$\gamma_n$', in the sense that on the horizontal segments
$\{v=\pm R\}$ one has $\zeta'(w)/\zeta(w)\ll \log^2|v|$.
One may require that $R_k\in(k,k+1)$, $k=1,2,\cdots$;
cf.\ the expansion of $(\zeta'/\zeta)(\cdot)$ and ways of
estimating this quotient in Titchmarsh \cite{Ti86}. One can
now use the standard majorants to show that for
$3/4<\sigma<3/2$,
\begin{align} & \label{eq:6.5}
\int_{L(c,B)}\Gamma(z-s)\frac{\zeta'(z)}{\zeta(z)}\,dz
\int_{c_1+iR}^{iR}\Gamma(w-s)\frac{\zeta'(w)}{\zeta(w)}\,
\cdot \notag \\ & \quad\cdot
\,M^\lambda(z+w-2s)\cos\{\pi(z-w)/2\}dw\to
0\quad\mbox{as}\;\;R\to \infty.
\end{align}
Similarly for the segment where $v=-R$. We summarize what
we have found so far:
\begin{proposition} \label{prop:6.1}
For $1<\sigma<3/2$ and $|\tau|<B$ the integral for $T^\lambda(s)$
admits a holomorphic decomposition $T^\lambda(s)
= U^\lambda_1(s)+U^\lambda_2(s)$, see $(\ref{eq:6.2})$, in which
both integrals $U^\lambda_j(s)$ define analytic functions for
$3/4<\sigma<3/2$.
\end{proposition}

\medskip
In a second step, inverting order of integration where
necessary, the $z$-paths $L(c,B)$ in the
integrals (\ref{eq:6.2}), (\ref{eq:6.3}) for $U^\lambda_1(s)$
and $U^\lambda_2(s)$ are moved to $L(0)$. Again using the
residue theorem, this results in the decomposition
\begin{align} \label{eq:6.6}
T^\lambda(s) &=
\int_{L(0)}\,\cdots\,dz\int_{L(0)}\,\cdots\,dw
+2\int_{L(0)}\,\Gamma(z-s)\frac{\zeta'(z)}{\zeta(z)}\,
J(z,s)dz \notag \\ &
+\bigg\{\frac{\zeta'(s)}{\zeta(s)}\,J(s,s) - \Gamma(1-s)J(1,s)+
\sum_{\rho'}\,\Gamma(\rho'-s)J(\rho',s)\bigg\}.
\end{align}
The new integrals will define holomorphic functions for
$0<\sigma<1$. In the case of the single integral involving
the sum over $\rho=\beta+i\gamma$ in $J(z,s)$, where $\beta$ might
be close to $1$, this can be seen by moving the integral
over to $L(-\eta)$ with variable $\eta\in(0,1)$. On the
interval $\{1/2\le s\le 3/4\}$ the integrals are $\cal{O}(1)$
as $\lambda\to\infty$.

We now turn to the {\it big residue} in the last line of
(\ref{eq:6.6}). With the aid of (\ref{eq:6.4}) it leads to
nine terms. Four of these combine into the three terms of
$\Sigma^\lambda(s)$ in (\ref{eq:3.1}); here the sum
$\Sigma^\lambda_2(s)$ of the double series has to be formed in a
suitable way as described below. Another term gives the
important constituent
\begin{equation} \label{eq:6.7}
U^\lambda_3(s)\stackrel{\mathrm{def}}{=}\Gamma^2(1-s)M^\lambda(2-2s).
\end{equation}
By Proposition \ref{prop:2.1} the
function $U^\lambda_3(s)$ is meromorphic for $0<\sigma<1$, with
just one pole, a first-order pole at $s=1/2$. Since
$M^\lambda(z)$ has residue $-(2\lambda/\pi)A^E$ at $z=1$,
expansion about $s=1/2$ gives
\begin{equation} \label{eq:6.8}
U^\lambda_3(s)=\frac{A^E\lambda}{s-1/2}+H^\lambda_2(s),
\end{equation}
where $A^E=\int_0^1 E(\nu)d\nu$ and $H^\lambda_2(s)$ is
holomorphic for $0<\sigma<1$. On the interval $\{1/2\le s\le
3/4\}$ one has $H^\lambda_2(s)=\cal{O}(\lambda\log\lambda)$ as $\lambda\to
\infty$.

The other four terms provided by the last line of
(\ref{eq:6.6}) combine to
\begin{align} \label{eq:6.9}
U^\lambda_4(s) &=
-2\Gamma(1-s)\frac{\zeta'(s)}{\zeta(s)}\,M^\lambda(1-s)\sin(\pi s/2)
\notag \\ & \quad -
2\Gamma(1-s)\sum_\rho\,\Gamma(\rho-s)M^\lambda(1+\rho-2s)
\sin(\pi\rho/2).
\end{align}
This function is meromorphic for $0<\sigma<1$ with poles
at zeta's complex zeros that cancel one another, and
further poles at the points $s=\rho/2$. Thus $U^\lambda_4(s)$
is holomorphic for $1/2\le\sigma<1$, and for $1/4<\sigma<1$ under
RH. For $s\searrow 1/2$ one has $U^\lambda_4(s)=\cal{O}(\lambda)$
as $\lambda\to \infty$.

To justify the application of the residue theorem one now
has to show that (\ref{eq:6.5}) remains valid if $L(c,B)$
is replaced by $L(0)$, and also that
$$\int_{1+iR}^{iR}\Gamma(z-s)\frac{\zeta'(z)}{\zeta(z)}\,
J(z,s)dz\to 0\quad\mbox{as}\;\;R=R_k\to
\infty.$$
This is no problem if $1/2<\sigma<3/2$. Similarly for the
segment $v=-R$.

Since $T^\lambda(s)$ is holomorphic for $\sigma>1/2$ by Theorem
\ref{the:5.1}, the discussion above implies that the
`big residue' in (\ref{eq:6.6}) represents a holomorphic
function for $1/2<\sigma<1$, provided the sum
over $\rho'$ is interpreted as
$$\lim_{R=R_k\to\infty}\,\sum_{|{\rm Im\,}\rho'|<R}\,
\Gamma(\rho'-s)J(\rho',s).$$
It follows that the function $\Sigma^\lambda(s)$ in (\ref{eq:3.1})
likewise represents a holomorphic function for
$1/2<\sigma<1$, provided the double sum $\Sigma^\lambda_2(s)$
involving $\rho$ and $\rho'$ is interpreted as a limit of
$\sum_{|{\rm Im\,}\rho'|<R}\,\sum_{\rho}$.
Under RH the double series is absolutely convergent; cf.\
Lemma \ref{lem:4.2}. Summarizing we have
\begin{theorem} \label{the:6.2}
For $s=\sigma+i\tau$ with $1/2<\sigma<1$ there is a
holomorphic decomposition
\begin{equation} \label{eq:6.10}
T^\lambda(s) = \frac{A^E\lambda}{s-1/2} + \Sigma^\lambda(s) +
H^\lambda_3(s),\quad A^E=\int_0^1 E(\nu)d\nu,
\end{equation}
where $\Sigma^\lambda(s)$ is given by $(\ref{eq:3.1})$ with
proper interpretation of the double sum. The function
$H^\lambda_3(s)$ has an analytic continuation to the strip
$\{1/2\le\sigma<1\}$, and to the strip $\{1/4<\sigma<1\}$ under
RH. On the interval $\{1/2\le s\le 3/4\}$ one has
$H^\lambda_3(s)=\cal{O}(\lambda\log\lambda)$ as $\lambda\to \infty$.
\end{theorem}

\setcounter{equation}{0}
\section{Proofs for the main results in Section \ref{sec:3}}
\label{sec:7}
\begin{proof}[Proof of Theorem \ref{the:3.1}] The
proof is obtained by combining Theorem \ref{the:5.2},
Lem\-ma \ref{lem:5.3} and Theorem \ref{the:6.2}. For
$1/2<\sigma<1$ these results give the following holomorphic
representations for the sum $V^\lambda(s)$:
\begin{align} \label{eq:7.1}
V^\lambda(s) &= 2\sum_{0<
2r\le\lambda}\,E(2r/\lambda)D_{2r}(s)
= T^\lambda(s)-D_0(s)+H^\lambda_4(s) \notag \\ &
= T^\lambda(s)-\frac{1/4}{(s-1/2)^2}+H^\lambda_5(s) \\ &=
-\frac{1/4}{(s-1/2)^2}+\frac{A^E\lambda}{s-1/2}+\Sigma^\lambda(s)
+H^\lambda_6(s).\notag
\end{align}
Here $A^E=\int_0^1 E(\nu)d\nu$ and $\Sigma^\lambda(s)$ is given by
(\ref{eq:3.1}) with suitable interpretation of the double
sum. The functions $H^\lambda_j(s)$ are holomorphic for
$1/2\le\sigma<1$, and for $1/4<\sigma<1$ under RH. For the final
line of (\ref{eq:3.2}) one applies (\ref{eq:7.1}) to
$V^1(s)=0$ and subtracts the result from (\ref{eq:7.1}) for
$V^\lambda(s)$.

Results on $H^\lambda_1(s)$ in Theorem \ref{the:5.2} and
$H^\lambda_3(s)$ in Theorem \ref{the:6.2} show that on the
interval $\{1/2\le s\le 3/4\}$ one has
$H^\lambda_6(s)=\cal{O}(\lambda\log\lambda)$ as $\lambda\to\infty$.
We will see below that the sum of the double series
$\Sigma^\lambda_2(s)$ in the definition of $\Sigma^\lambda(s)$ may be
formed as a limit of square partial sums.
\end{proof}
\noindent{\scshape Discussion of $\Sigma^\lambda(s)$.}
The part of the double sum $\Sigma^\lambda_2(s)$ in which
$\gamma={\rm Im}\,\rho$ and $\gamma'={\rm Im}\,\rho'$ have the
same sign defines a meromorphic function for $1/2<\sigma<1$
whose only poles occur at complex zeros of $\zeta(\cdot)$.
Indeed, for $s$ different from those zeros the series is
absolutely convergent; to verify this one may use an analog
for sums of the final part of Lemma \ref{lem:4.2}.
\begin{proposition} \label{prop:7.1}
The function $\Sigma^\lambda_2(s)$ can be obtained as a limit of
square partial sums:
\begin{align} & \Sigma^\lambda_2(s) =\notag \\ &
\lim_{R\to\infty}\,\sum_{|\gamma|,\,|\gamma'|<R}
\,\Gamma(\rho-s)\Gamma(\rho'-s)
M^\lambda(\rho+\rho'-2s)\cos\{\pi(\rho-\rho')/2\},\notag
\end{align}
where $R$ `stays away' from the numbers $\gamma_n$ as
described in Section \ref{sec:6}.
\end{proposition}
\begin{proof} By the preceding we may restrict ourselves to
the double sum $\Sigma^\lambda_3(s)$ in which $\gamma$ and $\gamma'$
have opposite sign; as shown in Section \ref{sec:6} it is
the limit of $\sum_{|\gamma'|<R}\,\sum_{\gamma;\,\gamma\ga'<0}$ for
suitable $R\to\infty$. In order to prove that one can use
square partial sums it will suffice to show that for
fixed $s$ (different from the numbers $\rho'$)
with $1/2<\sigma<1$,
$$\sum_{-R<\gamma'<0}\,\sum_{\gamma>R}\,\Gamma(\rho'-s)\Gamma(\rho-s)
M^\lambda(\rho+\rho'-2s)\cos\{\pi(\rho-\rho')/2\}\to 0$$
as $R\to\infty$. Using standard
majorants and setting $\gamma'=-\gamma''$ this will follow if we
prove
\begin{equation} \label{eq:7.2}
\sum_{0<\gamma''<R,\,\gamma>R}\,(\gamma\ga'')^{(1/2)-\sigma}
(\gamma-\gamma''+1)^{2\sigma-7/2}\to 0
\end{equation}
as $R\to\infty$. Now the number of zeta's zeros with
$T<\gamma\le T+1$ is $\cal{O}(\log T)$, cf.\ Titchmarsh
\cite{Ti86}. Thus for fixed $\sigma\in(1/2,1)$ the double sum
in (\ref{eq:7.2}) is majorized by
\begin{align} &
\int_2^R v^{(1/2)-\sigma}(\log v)dv\int_R^\infty
y^{(1/2)-\sigma}(y-v+1)^{2\sigma-7/2}(\log y)dy \notag \\ &
\ll \int_2^R v^{(1/2)-\sigma}(\log v)R^{(1/2)-\sigma}(\log R)
(R-v+1)^{2\sigma-5/2}dv \notag \\ & \ll
R^{(1/2)-\sigma}\log^2 R\int_2^R
v^{(1/2)-\sigma}(R-v+1)^{2\sigma-5/2}dv.\notag
\end{align}
In the last integral one may treat the $v$-intervals
$(1,R/2)$ and $(R/2,R)$ separately to obtain the final
majorant
$$(R^{-1/2}+R^{1-2\sigma})\log^2 R,\quad\mbox{which}\;\;\to
0\quad\mbox{as}\;\;R\to\infty.$$
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:3.2}]
The proof uses induction with respect to $m\ge 1$. The
induction hypothesis is that the PPC for pairs
$(p,\,p+2r)$ is known to hold for every $r<m$.

(i) Suppose that the PPC is also true for $r=m$. Then if
$\lambda$ is any given number in $(2m,2m+2]$, the PPC is true
for every $r<\lambda/2$. Hence for any smooth $E$, by
(\ref{eq:2.4}) the function
\begin{equation} \label{eq:7.3}
W^\lambda(s) \stackrel{\mathrm{def}}{=}
2\sum_{0<2r<\lambda}\,E(2r/\lambda)
\Big(D_{2r}(s)-\frac{C_{2r}}{s-1/2}\Big)
\end{equation}
has good (local pseudofunction) boundary behavior as
$\sigma\searrow 1/2$. Now by (\ref{eq:3.2})--(\ref{eq:3.4})
one has, for the present $\lambda$,
$$G^\lambda(s)=W^\lambda(s)-H^\lambda(s),$$
where $H^\lambda(s)$ is holomorphic for $1/2\le\sigma<1$. Hence
$G^\lambda(s)$ will also have good boundary behavior.

(ii) Conversely suppose that $G^\lambda(s)$ has good
boundary behavior as $\sigma\searrow 1/2$ for some smooth $E$
and some $\lambda\in(2m,2m+2]$. Then with this $\lambda$, the sum
$W^\lambda(s)$ in (\ref{eq:7.3}) has good boundary behavior. But
we know from the induction hypothesis that
$$2\sum_{0<2r<2m}\,E(2r/\lambda)
\Big(D_{2r}(s)-\frac{C_{2r}}{s-1/2}\Big)$$
has good boundary behavior, hence so does the difference
$$2E(2m/\lambda)\Big(D_{2m}(s)-\frac{C_{2m}}{s-1/2}\Big).$$
Since $E(2m/\lambda)\ne 0$ this implies the PPC for pairs
$(p,\,p+2m)$.
\end{proof}

\setcounter{equation}{0}
\section{Proof of Theorem \ref{the:3.4}}
\label{sec:8}
Assume RH and let $N(T)$ denote the number of zeta's zeros
$\rho=(1/2)+i\gamma$ with $0<\gamma\le T$. Setting $s=(1/2)+\delta$
with $0<\delta<1/4$ we will write $\Sigma^\lambda_0(\delta)
\stackrel{\mathrm{def}}{=}\Sigma^\lambda_3(s)$ as an integral.
Recall that $\gamma={\rm Im}\,\rho$ and $\gamma'={\rm Im}\,\rho'$
in $\Sigma^\lambda_3(s)$ have opposite sign. Thus it is convenient
to introduce
\begin{align} & \label{eq:8.1}
F^\lambda(y,v,\delta) = \\ &
\Gamma(iy-\delta)\Gamma(-iv-\delta)M^\lambda\{-2\delta+i(y-v)\}
\cosh\{\pi(y+v)/2\}.\notag
\end{align}
Since $F^\lambda(y,v,\delta)=F^\lambda(-v,-y,\delta)$ one finds that
\begin{equation} \label{eq:8.2}
\Sigma^\lambda_0(\delta)=\Sigma^\lambda_3(s)=
2\int\int_{y,\,v>2} F^\lambda(y,v,\delta)dN(y)dN(v).
\end{equation}
For the study of $F^\lambda(\cdot)$ we use Stirling's uniform
asymptotic formula for $|\arg z|<\pi-\varepsilon$ and $|z|>2$:
\begin{equation} \label{eq:8.3}
\log\Gamma(z)=(z-1/2)\log z-z+(1/2)\log(2\pi)+\cal{O}(1/|z|);
\end{equation}
cf.\ Whittaker and Watson \cite{WW27}. It shows that for
$|y|>2$
\begin{align}
\Gamma(iy-\delta) & =\sqrt{2\pi}e^{-\pi i({\rm
sgn}\,y)(1+2\delta)/4}|y|^{-\delta-1/2}
\big[1+\cal{O}(1/|y|)\big]\cdot\notag \\ &
\quad\cdot\exp\{iy\log |y|-iy-\pi|y|/2\}.\notag
\end{align}
This formula will be used also for $\Gamma(-iv-\delta)$. Thus by
(\ref{eq:8.1}), (\ref{eq:2.8}) and (\ref{eq:2.9}), for
$y,\,v>2$,
\begin{align} \label{eq:8.4}
& F^\lambda(y,v,\delta) =\pi(yv)^{-\delta-1/2}
e^{i(y\log y-y-v\log v +v)}\,\cdot \notag\\
& \;\;\cdot\lambda^{-2\delta+i(y-v)}M\{-2\delta+i(y-v)\}
\big[1+\cal{O}(1/y)+\cal{O}(1/v)\big] \\ &
\ll G^\lambda(y,v,\delta)\stackrel{\mathrm{def}}{=} \lambda^{-2\delta}
(yv)^{-\delta-1/2}(|y-v|+1)^{2\delta-7/2}.\notag
\end{align}

To complete the proof of Theorem \ref{the:3.4} we establish
\begin{proposition} \label{prop:8.1}
The pole-type behavior of $\Sigma^\lambda_0(\delta)=\Sigma^\lambda_3(s)$ as
$\delta\searrow 0$ is the same as that of the reduced function
\begin{align} \label{eq:8.5}
\Sigma^\lambda_4(s) = \Sigma^\lambda_*(\delta) &= 2\pi\int\int_{y,\,v>2;\,
|y-v|<y^{1/2}}\,
y^{-1-2\delta+i(y-v)}\,\cdot \notag \\ &\qquad\qquad\cdot
M^\lambda\{-2\delta+i(y-v)\}dN(y)dN(v).
\end{align}
\end{proposition}
\begin{proof}
In the discussion of the integral $I^\lambda$ of $F^\lambda$ one may
ignore the quantities $\cal{O}(1/y)$ and $\cal{O}(1/v)$ that
occur in (\ref{eq:8.4}); by Lemma \ref{lem:4.2} they lead
to bounded functions of $\delta$. Furthermore, it follows
from the majorization (\ref{eq:8.4}) that the integral
$I^\lambda_1$ of $|F^\lambda(y,v,\delta)|dN(y)dN(v)$ over the set
$\Omega_1$ where $y,\,v>2$ and $|y-v|\ge y^{1/4}$ is bounded on
the interval $\{0<\delta<1/4\}$. Indeed, for fixed $\lambda$,
\begin{align}
I^\lambda_1 &\ll \int\int_{\Omega_1}\,G^\lambda(y,v,\delta)
dN(y)dN(v)\notag\\ & \ll \int_2^\infty
y^{-\delta-1/2}(\log y)dy\int_{|v-y|\ge
y^{1/4}}\,(|v-y|+1)^{2\delta-7/2}(\log v)dv\notag \\ &\ll
\int_2^\infty y^{-\delta-1/2}(\log y)\cdot y^{(\delta/2)-5/8}(\log
y)dy. \notag
\end{align}

It follows that we may surely restrict ourselves to the part
$I^\lambda_2$ of the integral in (\ref{eq:8.2}) over the set
$\Omega_2$ where $y,\,v>2$ and $|y-v|<y^{1/2}$. On this set the
function
$$v^{-\delta-1/2}=y^{-\delta-1/2}\{1+(v-y)/y\}^{-\delta-1/2}=
y^{-\delta-1/2}+\cal{O}(y^{-\delta-1})$$
may be replaced by $y^{-\delta-1/2}$; by Lemma \ref{lem:4.2}
the error term gives rise to a bounded function of $\delta$. We
finally observe that on $\Omega_2$
$$y\log y-y-v\log v +v=\int_v^y(\log t)dt=(y-v)(\log y)
+\cal{O}\{|y-v|^2/y\},$$
hence
$$e^{i(y\log y-y-v\log v +v)}=y^{i(y-v)}\big[1+
\cal{O}\{|y-v|^2/y\}\big].$$
The contribution to $I^\lambda_2$ due to the final
$\cal{O}$-term is bounded on the interval $\{0<\delta<1/4\}$.
Thus as regards pole-type behavior, the function
$\Sigma^\lambda_0(\delta)$ can be reduced to $\Sigma^\lambda_*(\delta)$ or
$\Sigma^\lambda_4(s)$.
\end{proof}
\begin{remark} \label{rem:8.2}
The pole-type behavior of $\Sigma^\lambda_3(s)$ and $\Sigma^\lambda(s)$
as $s\searrow 1/2$ is also the same as that of the
symmetric sum
$$2\pi\sum_{\gamma,\,\gamma';\,|\gamma-\gamma'|<(\gamma\ga')^{1/6}}
(\gamma\ga')^{-s+i(\gamma-\gamma')/2}M^\lambda\{1-2s+i(\gamma-\gamma')\}.$$
\end{remark}

\setcounter{equation}{0}
\section{Pair correlation of zeta's zeros and\\
the conditional Theorem \ref{the:3.6}}
\label{sec:9}
Goldston, Pintz and Yildirim \cite{GPY05} have
shown conditionally that there are infinitely many prime
pairs $(p,\,p+2r)$ for some $r$ with $2r\le 16$. Their
proof used a hypothesis of Elliott and
Halberstam \cite{EH68} on (weighted) equidistribution of
primes in arithmetic progressions. Below we will discuss the
conditional Theorem \ref{the:3.6} which would imply that
there is an abundance of prime pairs for some difference
$2r$. The proof depends on Hypothesis
\ref{hyp:3.5}, which is supported by Theorem \ref{the:3.4}
and related aspects of the pair correlation of zeta's complex
zeros.

As an introduction we describe some results on the pair
correlation from the now extensive literature. In the
background was a comparison, under RH, of the fine
distribution of zeta's zeros $\rho=(1/2)+i\gamma$ to the
eigenvalue distribution of large unitary matrices; see the
LMS Lecture Notes vol.\ 322 \cite{LMS05}. Using an ingenious
computation, Montgomery \cite{Mo73} obtained the following
basic pair-correlation result, cf.\ Goldston and Montgomery
\cite{GM87}:
\begin{theorem} \label{the:9.1}
Assume RH and set $w(u)=4/(4+u^2)$ so that $w(0)=1$. Then for
$T\to\infty$, uniformly for $\alpha\in[0,1]$,
\begin{align} \label{eq:9.1}
F_w(\alpha,T) & \stackrel{\mathrm{def}}{=}
\frac{2\pi}{T\log T}\sum_{0<\gamma,\,\gamma'\le
T}\,e^{i\alpha(\gamma-\gamma')\log T}w(\gamma-\gamma')\notag\\
&=\{1+o(1)\}T^{-2\alpha}\log T+\alpha+o(1).
\end{align}
\end{theorem}
One may speculate that (\ref{eq:9.1}) holds for many other
weight functions $w$ with $w(0)=1$; cf.\  Hejhal \cite{He94},
Rudnick and Sarnak \cite{RS94}, \cite{RS96},
Bogomolny and Keating \cite{BK95}. Furthermore, Montgomery
used the PPC to support the conjecture that, uniformly
for $1\le\alpha\le C$,
\begin{equation} \label{eq:9.2}
F_w(\alpha,T)=1+o(1)\quad\mbox{as}\;\;T\to\infty.
\end{equation}
This conjecture would imply that almost all of zeta's
complex zeros are simple: $N_s(T)\sim N(T)\sim(T\log
T)/(2\pi)$. It also implies that the behavior of
$F_w(\alpha,T)$ for $\alpha\ge 1$ is determined largely by the
terms in the double sum of (\ref{eq:9.1}) for which
$\gamma'=\gamma$; the terms with $\gamma'\ne\gamma$ would essentially
cancel each other. To visualize the exponentials
$e^{i\alpha(\gamma-\gamma')\log T}$ on the unit circle, observe that
the mean spacing of zeta's zeros $(1/2)+i\gamma$ for
$\gamma$ near $T$ is approximately $2\pi/\log T$.

See also the subsequent work by Gallagher and Mueller
\cite{GaM78}, Heath-Brown \cite{HB82}, Gallagher
\cite{Ga85}, Goldston \cite{Go88}, Goldston and Gonek
\cite{GG90}, \cite{GG98}, Goldston, Gonek,
\"{O}zl\"{u}k and Snyder \cite{GGOS00}, Montgomery and
Soundararajan \cite{MS04}, Chan \cite{Cha03}, \cite{Cha04a},
\cite{Cha04b}, and Goldston \cite{Go05}.

\smallskip
Always assuming RH, it is interesting to compare the case
$\alpha=1$ of Montgomery's result and the case $\lambda=1$ of
Theorem \ref{the:3.4}. By (\ref{eq:9.1})
$$2\pi\sum_{0<\gamma,\,\gamma'\le
T}\,e^{i(\gamma-\gamma')\log T}w(\gamma-\gamma')\sim T\log
T\quad\mbox{as}\;\;T\to\infty,$$
while by (\ref{eq:7.1}) (since $V^1(s)=0$) and
Theorem \ref{the:3.4} (with $2s=1+\delta$)
$$2\pi\sum_{\gamma,\,\gamma';\,|\gamma-\gamma'|<\gamma^{1/2}}\,
\gamma^{-1-\delta+i(\gamma-\gamma')}M^\lambda\{-\delta
+i(\gamma-\gamma')\}\sim 1/\delta^2$$
as $\delta\searrow 0$. It appears that in first approximation,
the behavior of the second sum is also determined by the
terms with $\gamma'=\gamma$:
$$2\pi\int_2^\infty
y^{-1-\delta}M^\lambda(-\delta)dN(y)\sim\int_2^\infty y^{-1-\delta}(\log
y)dy\sim 1/\delta^2.$$

\begin{proof}[Support for Hypothesis \ref{hyp:3.5}]
Using Theorems \ref{the:3.1} and \ref{the:3.4} with
$s=(1/2)+\delta$, and writing $\Sigma^\lambda_4(s)=\Sigma^\lambda_*(\delta)$ as
in (\ref{eq:8.5}), we will now consider the difference
\begin{align} \label{eq:9.3}
\Sigma^\lambda_*(\delta)-\Sigma^1_*(\delta) &=  2\pi
\sum_{\gamma,\,\gamma';\,|\gamma-\gamma'|<\gamma^{1/2}}\,
\big\{\lambda^{-2\delta+i(\gamma-\gamma')}-1\big\}\cdot \notag \\ &
\qquad\qquad\cdot\gamma^{-1-2\delta+i(\gamma-\gamma')}M\{-2\delta
+i(\gamma-\gamma')\}.
\end{align}
For $\lambda=2$ it follows from Theorem
\ref{the:3.1} (since $V^2(s)=0$) that
$$\Sigma^2_*(\delta)-\Sigma^1_*(\delta)=-\frac{A^E}{\delta}+\cal{O}(1)
\quad\mbox{as}\;\;\delta\searrow 0.$$
Compared to the original sum $\Sigma^1_*(\delta)$, the general term
in (\ref{eq:9.3}) now contains an additional factor
$2^{-2\delta+i(\gamma-\gamma')}-1$. For small $\delta$ and $\gamma-\gamma'$,
this factor is like
$\{-2\delta+i(\gamma-\gamma')\}\log 2$. If one may ignore the
contribution due to larger $|\gamma-\gamma'|$, the effect of
the factor will be roughly
$$\{-2\delta(\log 2)/(4\delta^2)+\,\mbox{contribution of}\;\;
i(\gamma-\gamma')(\log 2)/(4\delta^2).$$
We know that the new pole is $-A^E/\delta$, hence the second
contribution must also result in a first order pole with
modest residue.

In the case of general $\lambda$ the effect of the factor
$$\lambda^{-2\delta+i(\gamma-\gamma')}-1\approx \{-2\delta+i(\gamma-\gamma')\}\log
\lambda$$
might well be a first order pole with residue of order
$\log\lambda$; cf.\ also (\ref{eq:2.6}) and (\ref{eq:1.6}). Thus
Hypothesis
\ref{hyp:3.5} appears to be  plausible.
\end{proof}
\begin{proof}[Proof of Theorem \ref{the:3.6}] For given
$\varepsilon\in(0,1)$ we form a smooth sieving function
$E^\lambda(\nu)=E(\nu/\lambda)$ (as in Section \ref{sec:2}) such
that
\begin{equation} \label{eq:9.4}
A^E=\int_0^1 E(\nu)d\nu > 1-\varepsilon/3.
\end{equation}
For $\lambda>2$ and $s\in(1/2,1)$ we now use the
final representation for $V^\lambda(s)$ in Theorem \ref{the:3.1}:
\begin{align} \label{eq:9.5}
V^\lambda(s) &= 2\sum_{0<2r\le\lambda}\,E(2r/\lambda)D_{2r}(s)
\notag \\ &= \frac{A^E(\lambda-1)}{s-1/2}+\Sigma^\lambda(s)-\Sigma^1(s)+
H^\lambda(s),
\end{align}
where $H^\lambda(s)$ is holomorphic for $1/2\le\sigma<1$. Thus by
Hypothesis \ref{hyp:3.5}
\begin{equation} \label{eq:9.6}
\limsup_{s\searrow
1/2}\,(s-1/2)V^\lambda(s)=A^E(\lambda-1)+\omega(\lambda),
\end{equation}
where $\omega(\lambda=o(\lambda)$ as $\lambda\to\infty$. Hence by
(\ref{eq:9.4}), the right-hand side of (\ref{eq:9.6}) will
be greater than $(1-\varepsilon/2)\lambda$ for all sufficiently large
$\lambda$. We choose the smallest even positive integer
$\lambda=2m$ for which this is so. Since $0\le E(\nu)\le 1$,
it then follows from (\ref{eq:9.5}) that
\begin{equation} \label{eq:9.7}
\limsup_{s\searrow
1/2}\,(s-1/2)\sum_{r\le m}\,D_{2r}(s)>(1-\varepsilon/2)m.
\end{equation}
We will show that this implies
\begin{equation} \label{eq:9.8}
\limsup_{x\to\infty}\,(1/x)\sum_{r\le m}\,\psi_{2r}(x)>
C=(2-\varepsilon)m.
\end{equation}

Suppose to the contrary that
$(1/x)\sum_{r\le m}\,\psi_{2r}(x)\le C$ for all $x\ge x_0\ge
1$. Now by (\ref{eq:2.3}) and (\ref{eq:2.1}) for
$s\in(1/2,1)$,
\begin{align}
D_{2r}(s) &= \int_1^\infty x^{-s}(x+2r)^{-s}d\psi_{2r}(x)
\le \int_{x_0}^\infty x^{-2s}d\psi_{2r}(x)+\cal{O}(1)
\notag \\ &\le 2s\int_{x_0}^\infty
x^{-2s-1}\psi_{2r}(x)dx+\cal{O}(1).\notag
\end{align}
Hence it would follow that
\begin{align} & \label{eq:9.9}
\sum_{r\le m}\,D_{2r}(s)\le 2s\int_{x_0}^\infty
x^{-2s-1}\sum_{r\le m}\,\psi_{2r}(x)dx+\cal{O}(1) \notag\\
&\le 2s\int_{x_0}^\infty Cx^{-2s}dx+\cal{O}(1)
\le 2sC/(2s-1)+\cal{O}(1).
\end{align}
As a result the upper residue in (\ref{eq:9.7}) would
be $\le C/2$, hence $\le(1-\varepsilon/2)m$. This contradiction
proves (\ref{eq:9.8}).

In order to pass from (\ref{eq:9.8}) to (\ref{eq:3.7}) one
may appeal to the discussion in Section \ref{sec:2}.
\end{proof}

\setcounter{equation}{0}
\section{Positivity of sums $\Sigma^\lambda_2(s)$ and \\
conditional abundance of prime pairs} \label{sec:10}
We will verify the positivity of the double sums
$\Sigma^\lambda_2(s)$ in (\ref{eq:3.1}) for $1/2<s<1$ when $\hat
E^\lambda(t)\ge 0$.
\begin{proof}[Proof of Proposition \ref{prop:3.7}]
It will be convenient to replace $\rho'$ in the double sum
$\Sigma^\lambda_2(s)$ by $\overline\rho'$. Set
$\Omega_R(t,s)=\Omega'_R(t,s)+\Omega''_R(t,s)$, where
$$ \Omega'_R(t,s)= \sum_{|{\rm Im}\,\rho|,\,|{\rm
Im}\,\rho'|<R}\,\Gamma(\rho-s)
\Gamma(\overline\rho'-s)t^{2s-\rho-\overline\rho'}
\cos(\pi\rho/2)\cos(\pi\overline\rho'/2),$$
and $\Omega''_R(t,s)$ is the corresponding function with $\sin$
instead of $\cos$. Then
$$\Omega'_R(t,s)=\bigg|\sum_{|{\rm Im}\,\rho|<R}
\Gamma(\rho-s)t^{s-\rho}\cos(\pi\rho/2)\bigg|^2\ge 0,$$
and similarly for $\Omega''_R(t,s)$. Hence by Proposition
\ref{prop:7.1} and (\ref{eq:2.7})
$$\Sigma^\lambda_2(s)=\frac{1}{\pi}\lim_{R\to\infty}\,\int_0^\infty
\hat E^\lambda(t)\Omega_R(t,s)dt\ge 0.$$
\end{proof}

One may use Proposition \ref{prop:3.7} to derive another
conditional abundance result:
\begin{theorem} \label{the:10.2}
Suppose that for certain positive integers
$m_1<m_2<\cdots<m_k$ there are a constant $c>0$ and a
sequence $S$ of numbers $\lambda\to \infty$, such that for
$\lambda\in S$ and sufficiently large $x$,
say $x\ge x_1=x_1(\lambda)$ with $\log x_1(\lambda)=o(\lambda)$, one has
\begin{equation} \label{eq:10.1}
\frac{1}{k}\,\sum_{j=1}^k\,\pi_{2m_j}(x)\ge
c\cdot\frac{2}{\lambda}\sum_{0<2r\le\lambda}\,\pi_{2r}(x).
\end{equation}
Then
\begin{equation} \label{eq:10.2}
\limsup_{x\to\infty}\,\frac{1}{k}\,\sum_{j=1}^k\,
\frac{\pi_{2m_j}(x)}{x/\log^2 x}\ge c.
\end{equation}
\end{theorem}
There is both heuristic and numerical support for the
hypothesis of the theorem. The proof below makes use of the
sieving function $E^\lambda(\nu)=E^\lambda_F(\nu)$. Although it does
not satisfy the smoothness requirement imposed in Section
\ref{sec:2}, one can show that it may be used anyway; it
gives a better result here than $E^\lambda_J$. We plan to
return to the details later; cf.\ also \cite{Ko07}.
\begin{proof}[Brief indication of the proof] It suffices to
treat the case $k=1$, the general case being similar; we
write $m_k=m$. Now Theorem \ref{the:3.1} with $E=E_F$, so that
$A^E=1/2$, the decomposition $\Sigma^\lambda(s)=\Sigma^\lambda_1(s)+\Sigma^\lambda_2(s)$
and Proposition \ref{prop:3.7} imply the following inequality
for $1/2<s<3/4$:
$$ 2\sum_{0<2r\le\lambda}\,E(2r/\lambda)
D_{2r}(s) \ge -\frac{1/4}{(s-1/2)^2}+
\frac{\lambda/2}{s-1/2}+\Sigma^\lambda_1(s)-\cal{O}(\lambda\log\lambda).$$
Here $0\le E(2r/\lambda)\le 1$ and the sum $\Sigma^\lambda_1(s)$ of the
first two terms in (\ref{eq:3.1}) is $\cal{O}(\lambda^{1/2})$.
Setting $s-1/2=\delta$ it follows that
\begin{equation} \label{eq:10.3}
\frac{2}{\lambda}\sum_{0<2r\le\lambda}\,\delta
D_{2r}\{(1/2)+\delta\}\ge
\frac{1}{2}-\frac{1}{4\lambda\delta}-\cal{O}(\delta\log\lambda).
\end{equation}
Combining this with the hypothesis of the theorem,
appropriate estimates show that
\begin{equation} \label{eq:10.4}
\delta D_{2m}\{(1/2)+\delta\}
\ge c\bigg(\frac{1}{2}-\frac{1}{4\lambda\delta}\bigg)
-\cal{O}\{\delta\log(x_1(\lambda)\}.
\end{equation}
For given $\varepsilon\in(0,1/2)$ we now choose $\lambda\to\infty$ in $S$ and
$\delta\searrow 0$ in $(0,1/4)$ such that $1/(4\lambda\delta)=\varepsilon/2$. Since
$\log x_1(\lambda)=o(\lambda)$ one may conclude that
\begin{equation} \label{eq:10.5}
\limsup_{\delta\searrow 0}\,\delta
D_{2m}\{(1/2)+\delta\}\ge(1-\varepsilon)c/2.
\end{equation}
From here on one may argue as in the proof of Theorem
\ref{the:3.6}.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Q: Array to Return Year End Balances For The Past Five Years - (C#) I am trying to return only the year end information (December 31) for the past five years and display that information in a DataGrid. This does not include the current year.
My program is currently displaying month's end information from the previous year up to the last day of the most recent month in the current year (Example: The current month is May 2011 and the information being displayed in my DataGrid shows the data stored on Jan. 31, 2010; Feb. 28, 2010; Mar. 31, 2010....Dec. 31, 2010; Jan. 31, 2011; Feb. 28, 2011; Mar. 31, 2011; Apr. 30, 2011).
It also adds a totals row after Dec. 31, 2010 and another after Apr. 30, 2011. These two totals rows will not be needed in the new DataGrid.
Below is a snippet of the current code which is returning the months end information.
Please let me know if I left any information out or am not clear on my question.
I apologize in advance for such a rookie question, it has just been driving me crazy the past couple days that I cannot figure this out.
protected DataSet DS;
protected DataRow dr, dNew;

private void GetInfo()
{

}

private double getDbl(string columnName)
{
    if (dr[columnName] != DBNull.Value)
        return Double.Parse(dr[columnName].ToString());
    return 0;
}

private void Calculate()
{
    double cg, na, n, ka, nka, kaa, tca;
    double tmr, to, gpo;
    int year, rowcnt, rowindx, months;

    months = -1;
    rowindx = 0;
    cg = 0;
    na = 0;
    n = 0;
    ka = 0;
    nka = 0;
    kaa = 0;
    tca = 0;
    tmr = 0;
    tor = 0;
    gpo = 0;

    rowcnt = DS.Tables[0].Rows.Count;

    if (rowcnt > 0) year = Int32.Parse(DS.Tables[0].Rows[0]["cy"].ToString());
    else year = 0;

    for (int i = 0; i < rowcnt; i++)
    {
        months++;
        dr = DS.Tables[0].Rows[i]

        if (year != Int32.Parse(dr["cy"].ToString())
        {
            dNew = DS.Tables[0].NewRow();
            dNew["ind"] = rowindx;
            dNew["cg"] = cg;
            dNew["na"] = na;
            dNew["n"] = n;
            dNew["ka"] = ka / months;
            dNew["nka"] = nka / months;
            dNew["kaa"] = kaa / months;
            dNew["tca"] = tca / months;
            dNew["tmr"] = tmr / months;
            dNew["tor"] = tor;
            dNew["gpo"] = gpo / months;

            cg = 0;
            na = 0;
            n = 0;
            ka = 0;
            nka = 0;
            kaa = 0;
            tca = 0;
            tmr = 0;
            tor = 0;
            gpo = 0;
            months = 1;
            year = Int32.Parse(dr["cy"].ToString());
            rowindx++;

            DS.Tables[0].Rows.Add(dNew);

            dNew = DS.Tables[0].NewRow();
            dNew["ind"] = rowindx;
            rowindx++;
            DS.Tables[0].Rows.Add(dNew);
        }

        cg += getDbl("cg");
        na += getDbl("na");
        n += getDbl("n");
        ka += getDbl("ka");
        nka += getDbl("nka");
        kaa += getDbl("kaa");
        tca += getDbl("tca");
        tmr += getDbl("tmr");
        tor += getDbl("tor");
        gpo += getDbl("gpo");
        dr["ind"] = rowindx;
        rowindx++;
    }

    dNew = DS.Tables[0].NewRow();
    dNew["ind"] = rowindx;
    dNew["cg"] = cg;
    dNew["na"] = na;
         dNew["n"] = naum;
         dNew["ka"] = ka / months;
         dNew["nka"] = nka / months;
         dNew["kaa"] = kaa / months;
         dNew["tca"] = tca / months;
         dNew["tmr"] = tmr / months;
         dNew["tor"] = tor;
         dNew["gpo"] = gpo / months;
         DS.Tables[0].Rows.Add(dNew);

      DS.Tables[0].DefaultView.Sort = "ind";
}

A: Have you tried limiting the dataset query to just the year the you want?

redpajama_set_name: RedPajamaStackExchange


text:
The psychology of genocide attempts to explain genocide by means of psychology. Psychology of genocide aims to explain the preconditions of genocide and why some people become genocide perpetrators while others are bystanders or rescuers.

Preconditions
Psychologists have agreed that specific prerequisites stimulate the act of genocide:

 Staub's model of frustration elicits that the depletion of basic human needs, such as economic stability, sparks collective frustration.
 This leads to introducing a scapegoat who is misconstrued as the root source of their detrimental life conditions and the in-group are depicted as victims. The selection of a scapegoat follows a process that results in the total domination of the in-group and devaluation of the chosen scapegoat.
 Pre-existing differences between the ingroup and the target group, such as ethnic or religious contrasts, radically shift to become immensely damaging to the livelihood of the in-group. For instance, the 2 million Armenians living in Ottoman Turkey were marginalised for their belief in Christianity.
 The subsequent stage is that the perpetrators create an ideology that emphasises that a utopian future can become a soon reality. This process involves exploiting the human innate fear mechanism by highlighting that the sole means to survive is to systematically eradicate the scapegoat. The leading perpetrators begin to construct a mythological explanation that aims to eliminate empathy and compassion directed towards the target group. The use of Nazi propaganda is a sufficient example as the Minister of Propaganda, Joseph Goebbels (1933–45), showcased a myriad of films dehumanising Jews and portraying them as a lethal virus. Political leaders aim to occupy the moral consciousness of wider society and force this illusion of unanimity to gain total state control.

Perpetrators
Perpetrators are the individuals who carry out, facilitate or instruct the annihilation of a specific group. Psychologists have historically debated whether dispositional or situational variables hold greater validity as explanations for the behaviour of perpetrators.

Dispositional variables
Adorno postulated that possessing an authoritarian personality is the most integral cause of perpetrators' violence. He concluded that the three integral components of authoritarianism are conventionalism, submission to authority and aggression. Perpetrators also share the behaviour of killing without remorse which enables them to repeat more violent atrocities. Adorno's findings were derived from the 30 item F scale, which measured the extent to which participants agreed with authoritarian statements. One of the items is "Respect for authority is the most important virtue children should learn".

Situational variables
Milgram contends that obedience plays a significant role in transforming ordinary humans into transgressive perpetrators. His study measured the degree to which participants would administer shocks to learners just because the experimenter instructed them to do so. He found that, due to the effects of probing by the experimenter, 65% of participants obeyed instructions to the highest level (450 volts). Therefore, Milgram concluded that it is the result of precise situational arrangements which succeeded in muting perpetrators' inner moral conflict.

Solomon also hypothesised that the power of the situation constitutes the paramount reasoning for the behaviour of perpetrators. His research concluded that desensitisation directly implies the ongoing exposure to violence and bloodshed. The exposure greatly reduces their normative restraints (process A) and is substituted for norms accepting of mass brutalities (process B). Research has also shown a potential pleasure of aggression which can explain perpetrators' excessive torture and violence. For instance, Simpson, a soldier in the 1968 My Lai genocide, described cutting off the hands and tongues of his 25 victims. Later theorists concluded the divide between situational and dispositional variables is a false dichotomy as the power of the situation can result in a perpetual shift in an individual's personality.

Bystanders
Bystanders are individuals who remain passive and silent when witnessing the ethnic cleansing of a target group. Bystanders have also been regarded as semi active as many freely accept the benefits of being a member of the in-group whilst actively avoiding the victims, such as companies firing Jewish employees.

Internal bystanders
According to Zilmer and Harrower, bystanders are characterised as ambient, which is defined as individuals who lack sufficient emotional development and must rely on others for guidance. They also have lower levels of moral development, which leads to a more compliant and submissive personality. The same study found that a critical justification for limited emotional development is bystanders' failing attachments with their primary caregiver, who becomes instrumental in foreshadowing their apathetic behaviour. In McFarland-Icke's study of Nurses in Nazi Germany, she concluded that the lack of resistance to perpetrators results from the bystanders' inability to engage in higher-order processes such as deductive reasoning and logic.

External bystanders
International bodies and other nations have also been labelled as external bystanders due to their unwillingness to intervene in another state's domestic political issue. The fear of economic stagnation and limited trade interactions was critical for nations not to involve themselves in genocides. For instance, before the 1994 Rwandan genocide took place, the commander of the United Nations peacekeepers received information about the upcoming massacre but was instructed by his seniors to keep the information concealed. The absence of economic support aided in the deaths of approximately 700,000 Tutsis in 3 months.

Rescuers
Rescuers are individuals who actively pursue helping the victims survive by providing shelter, protection or a means of escape.

Identity
Rescuers are identified as having internalised empathy and moral values, which serve as a diametric contrast to the growing presence of the perpetrators' ideologies. Theorists have also claimed that a strong sense of individuality is a critical force in driving rescuers' behaviour. This means that they do not conform to the majority's views and reject movement towards enculturation. Historian Christopher Browning discovered that an estimated 10-20% of Nazi soldiers evaded killing Jews due to their empathy and belief in individual choice.

Socialisation
Oliner's observations highlighted that the determining factor for the behaviour of rescuers is the role of the family. He found that mothers of rescuers transferred healthier moral competence and independence goals compared to mothers of non-rescuers. Rescuer families also embodied other codes of ethics, such as valuing collective responsibility and egalitarianism, irrespective of one's ethnicity or beliefs.

See also

 Milgram experiment
 Moral disengagement
 Moral psychology
 Holocaust denial

References

Further reading

Genocide
Applied psychology

redpajama_set_name: RedPajamaWikipedia


Fantastically wealthy GCC Giant, Saudi Arabia has much to offer those looking for Jobs in Oil and Gas, as well as other burgeoning sectors such as Telecoms, Banking and Finance.
Saudi Arabia is the largest of all of the Arab countries in the Middle East and is widely seen as center of the region both politically and culturally. Its cultural significance is tremendous as the region is predominantly Islamic and Saudi Arabia is home to the two holiest places in Islam, Mecca and Medina. From a political standpoint it is a major player in the Organization of Petroleum Exporting Countries (OPEC) and is a key ally in the region for the United States.
The current Kingdom of Saudi Arabia (the country's official name) was proclaimed in 1932 by Abdul-Aziz bin Saud, although its roots can be traced back more than 250 years. Between its history and its modern global importance, it is not surprising that Saudi Arabia is a desirable place for many people to work and live.
The population of Saudi Arabia is about 28 million people, with its overall size in area about 830,000 square miles. Much of this land is covered by desert, but along the coastal regions and the Jabal al-Hejaz Mountains the landscapes are much more forgiving and comfortable. Many Saudis flock to these more comfortable areas for their vacations, especially during the extremely hot summer months when the desert landscapes are sizzling with high heat.
The economy of Saudi Arabia is based predominantly on oil exports; by some estimates their petroleum production accounts for as much as 75% of their entire economy. In recent years the government has worked toward increasing the diversity of the economy by privatizing key sectors such as telecommunications and electricity, and also by launching a massive new city project along its western coastline. Called King Abdullah Economic City, this project is designed to promote more port activity and tourism, as well as more activity in education, finance, pharmaceuticals, and scientific research.
If you want to live and work in one of the richest countries in the world and be part of its dynamic economic changes, diverse culture and hot climate - Saudi Arabia is the place you want to go to find your new job.


text:
Burggräf oder Burggraef ist der Familienname folgender Personen:

 Eric Burggräf (* 1999), deutscher Volleyballspieler
 Joseph Burggraef (1876–1942), deutscher Landrat
 Peter Burggräf (* 1980), deutscher Ingenieurwissenschaftler und Hochschullehrer


Palladia King Headboard Select Cherry By Sauder.
A headboard is commonly one of the last pieces of furniture a homeowner purchases but it can make the biggest difference in the bedroom! This dark wood headboard is the perfect finishing touch for any bedroom.
Features include: Attaches to king size bed.


"Speak to dreamers in the language of dreams"
The Buddha said this world is like a dream or a conjuration. Therefore, to be comprehensible, his teachings must necessarily express themselves in a way which makes sense in each person's dream. Thus much of popular early Buddhism was taught using the Indo-European world view widespread at the time, in which, living atop the highest peak and ruling other deities was a sky god, known as Dyaus by the Indians, as Zeus by the Greeks and later as Jupiter (Dies Pater) by the Romans. Such concepts had probably spread from the Asiatic steppe, westwards to Greece and eastwards to India, with the invading Aryans, some time in the second millennium BCE. The sacred mountain was Olympus for some, Sumeru for others. Upon it, wielding a thunderbolt, the Lord of Heaven controlled the weather and repulsed attacks from demi-gods.
This early belief became extended to consider our world no longer as unique but as just one of a group of a billion similar world systems, each based around its own central, four-sided mountain, each face of which was made of a differently-coloured precious substance. Indians thought of their land, Jambudvipa, as being a trapezoidal continent to the south of the sacred mountain, opposite its lapis lazuli slope. To either side of it lay small sub-continents of similar shape. Other continents, flanked by sub-continents, lay opposite the other faces of Mt Sumeru: a semi-circular one to the east, opposite the crystal slope, a round one to the west, opposite the ruby slope and a square one to the north, opposite the emerald slope. In invisible worlds high above the summit of Mt. Meru, one above another, were the realms of the various classes of gods whereas in recesses far beneath the earth were the hells and lower abodes.
How relatively true were these primitive ideas was of secondary importance for Buddhists. What mattered was the fact that they were deeply ingrained in the psychology of millions of beings. If teachings based upon such a world view could enable someone to acquire the tools of meditation and clear analytical inspection, what matter? The true nature of reality would eventually become apparent through vivid first-hand knowledge, above and beyond all inherited conventional beliefs. Thus no ethical problem was seen in mobilising the myths and illusions of the day, as long as they set people on the path to wisdom.
On its deeper levels, Buddhism throws a bright light on the subjectivity of all experience. It reveals, with great pragmatism, the impossibility of establishing any ultimate objective reality and explains, as a consequence, that there are as many subjective worlds as there are sentient beings. Each moves through life in a completely unique universe fashioned by his, her or its preconceptions, due to karma. It is as though we live in parallel dreams. Furthermore, as one's awareness and mental clarity develops, many subconscious mental barriers fall away. In the newfound purity, the world around manifests to the senses very differently: there is 'a new heaven and a new earth'.
Surrounding Mount Sumeru are seven ranges of golden mountains, each separated by lakes of pure water of eight special attributes. These are reputed to be rich in precious gems, belonging to the serpentine naga spirits who inhabit them. Mount Sumeru has four large steps at its base and is unusual insofar as it tapers outwards to a flat, square summit rather than inwards to a peak. On the summit is the palace of Indra (who replaced Dyaus), Lord of the Heavens, surrounded by gardens and wonders. In space above, the sun and moon are themselves celestial palaces, as are the stars. Then, in layer after layer, one above another and interspersed with rainbow-hued celestial clouds, are first the seventeen realms of the form gods and above them the four realms of the formless gods.
The doctrine of karma explains the world around us to be the product of past actions, both personal and collective. The Kalachakra teachings describe cycles and tides of time, as humanity's karma carries it from age to age. Some ages - the results of much common goodness - are prosperous and peaceful with bountiful, healthy crops and longevity. Other darker ages - brought about by much evil - are riddled with disease, dishonesty, danger and a poisoned environment in which the lifespan is short. Unusual karmas produce unusual results and some worlds are said to be totally different from anything we could ever imagine. Furthermore, the tantras makes it clear that world views change as the centuries roll by. Within an endless series of parallel universes, we migrate from one to another much like actors appearing first on one television channel, then another, in quite different realities.
"Awaken to true reality, the vajra citadel of your mind"
Whereas the doctrine of karma deals with the cosmology of the exoteric world, the various tantric mandalas portray the inner, esoteric world which manifests to the mind of the experienced meditator. However, the notion of mandala itself is a very simple and universal one, being any focus on a central entity and the factors which usually accompany it. It could be used to describe something as banal, for instance, as the arrival of mains water in a remote desert village. That one central fact will bring with it the sale of bathroom items, a change in habits as people wash more frequently, cook more hygienically, make irrigable fields and so on and so forth. Mandala is a recognition that nothing exists all by itself but comes hand-in-hand with other things, and has inexorable implications and repercussions.
The sacred aspect of the mandala principle concerns the various facets of cosmic purity, which manifest as specific buddhas in their respective pure lands. Each of these buddhas is the symbolic embodiment of a particular quality of the primordial purity. Buddha Amitabha, for instance, displays its power to help the dying. Each main buddha is accompanied by an entourage relative to that buddha's activity. On this level, mandalas are maps of the awakened psyche; the geometry of enlightenment.
Mind's hidden treasures manifest themselves as mandalas in visions to great Buddhist saints and in the mandala they symbolise themselves through material qualities. Buddhas are 'seen', by the eye of mind, wearing raiments of celestial cloth and ornaments made by the gods. Their palaces of light seem to be made of sublime jewels, such as rubies, turquoises and pearls, and other precious substances. The paradisiacal gardens have pools of water having eight special qualities, being of right temperature, pure, sweet, soft, cleansing, bringing harmony, banishing hunger and thirst and satiating all the senses.
These buddha realms should not be confused with the realms of worldly gods, which are also full of splendours. They have radically different natures. One scripture even says that they do not even have one atom in common. Deva realms are the transient, illusory fruit of good karma. The enlightened mandala are the spontaneous manifestation of eternity: natural, archetypal symbols, generated by mind itself, of true reality. In particular, they represent the five primordial wisdoms. The number five is, in fact, very important in mandalas, as the majority represent a centre point and four sides. Thus most mandalas are geometrically square, contained within an outer circle. This is a two-dimensional way of representing a three-dimensional experience, in which buddhas are seen in a four-sided palace within an outer sphere. Other mandalas, especially wrathful ones, are triangular.
Every detail of the mandala is full of meaning. The outer sphere of interlaced vajras, which completely protects and hides its content from all but the initiate, represents the changeless and invincible realm of pure truth. Within this sphere, all is arranged according to the four cardinal directions. The central celestial palace has four portals, the main one to the east. In tantra, 'east' is whichever direction the yogin faces and is independent of the sun movements. In the heart of the palace, the most sacred spot, resides the particular buddha, like a king. Around him, in successive circles, are his entourage, compared to chief ministers, lesser ministers, servants, messengers and so forth, the less sacred being the furthest removed from the centre.
The entourage and the palace itself can be very detailed and composed of hosts of buddhas, bodhisattvas, attendant gods and goddesses. The mandala can also be very simple: just a few symbolic letters or a single central buddha flanked by two bodhisattvas, and no palace. In the more complex versions, the minutiae have profound meaning. The palace wall, for instance, can have five layers, representing the five wisdoms which are the true 'abode' of the enlightened mind. The palace floor is differently coloured in each of the cardinal directions, representing the infinite love, compassion, sympathetic joy and impartiality which form enlightenment's basis. Each detail tells its story.
A mandala is drawn on consecrated ground to create the sacred space used by a guru to bestow empowerment upon a disciple. This can also be set out on a table in the temple and hidden behind curtains or screens until the appropriate moment of the empowerment ceremony, when it is revealed to the initiate. Some temples have intricately-carved, three-dimensional mandala portraying a particular buddha and entourage. In others, buddhas and protectors are more simply represented by symbolic objects. Sometimes mandalas are painstakingly made of coloured sand, which is poured into a river once the ceremony completed.
After initiation, more permanent mandalas painted on a thangka or as a mural help the disciple to become familiar with what must be visualised in the creative stage of tantric practice. These paintings also serve as teaching aids for elaborating the detailed meaning of that particular buddha. After months or years of practice, the mandala will shift from being an external symbol to being an internal presence, becoming a familiar place in the meditator's imagination. Given enough time and devotion, this imagined mandala will magnetise the real, primordial mandala, locked within his or her subconscious, into replacing the limited mental creation and manifesting in all its breath-taking glory.
Be they squat domes of earth or gilded pagoda towers, stupas are one of the most striking features of Buddhist lands. Originally funeral mounds housing relics of the Buddha and his disciples, they have evolved into an elaborate architectural expression of the mandala. Unlike churches, mosques and temples, their prime function is not to house the disciple within a sacred space but to place a model of sacred order within the chaos of the worldly landscape.
Stupas are axes of power. Whatever their form, they are always symmetrical, aligned with points of the compass and radiating their geometric shapes around an all-important central axis, known as the 'life tree'. Although each shape and part of the stupa is charged with intentional significance - and can further lend itself to all sorts of symbolic interpretation - its prime importance for Buddhists lies in the relics it houses. Just before his passing, the Buddha gave instructions on the making of stupas and indicated that their construction and veneration would be mainly the task of the lay community. Monks, by contrast, should concentrate on their meditation and study but could, as time went by, keep the remains of their most saintly leaders enshrined in stupas in the monasteries, as a source of inspiration.
Following the funeral rites held for the Buddha and the pleas from each neighbouring kingdom to have some relics to enshrine, his remains were divided and preserved in eight stupas: at Rajgir, Vaisali, Kapilavastu, Allakappa, Ramagrama, Vethadipa, Pava and Kusinagara. Other non-corporeal relics from the Buddha's life, such as his hair, alms bowl, walking-staff, bathing-robe and so forth, also became objects of veneration.
Furthermore, stupas were built without his relics on the most sacred sites of his life, such as Bodh Gaya, where he attained enlightenment.
There remain but traces of the oldest stupas, dating back to the fifth century BCE. Still intact is the great stupa at Sanchi, built in brick by King Asoka in the third century BCE. Railings and gateways were added a century later. Typical of the earlier structures, it is a hemispherical dome, reminiscent of a funeral mound, truncated at the top and placed over a lofty terrace. Above the dome is a pavilion (harmika) above which is the parasol, symbolising the majesty of the Buddha's doctrine. In stupas at Sanchi, which is the site of the most extensive Buddhist remains in India, are relics of the Buddha's two main disciples, Sariputra and Mahamaudhgalyana.
constructed in such style at Nalanda and other places from the end of the Pala period and throughout the Gupta period. As it spread east to Nepal, China and Japan, the stupa took on more and more the form of a tower becoming, in its most extreme, the multi-tiered pagodas of the far East. However, in the south of India the original round mound continued to be built in such places as Amaravati. One of the two main stupas of Nepal - that at Bodhnath - retained this older style too.
bove the three main steps of the terrace, representing the three refuges of Buddha, dharma and sangha, appeared four steps representing infinite love, compassion, joy and impartiality. The terrace continued to evolve, reaching its ultimate artistic expression in the Nepali and Tibetan style where, after receding inwards it again expands outwards, like Mt Sumeru. Indeed, the stupa does represent this axis mundi and the dome and pavillion which surmounts it is analogous to the palace of the 'Lord of All'.
Symbolically, the stupa represents many things. Primarily it is a form representing formless dharmadhatu: the expanse of truth, the realm of voidness. One of many ways of explaining some of its symbolism is the following. The cubic base, with its terraces and steps, represents the earth element, the wisdom of sameness and the jewel family of beings. The mound or 'treasure vase' represents the water element, the mirror-like wisdom and the vajra family of beings and is particularly associated with the seven factors of enlightenment: mindfulness, awareness, diligence, joy, tranquillity, absorption and equanimity. The pavilion and spire together represent the fire element, discerning wisdom and the lotus family. It represents the eightfold path of right view, contemplation, speech, action, livelihood, effort, mindfulness and samadhi. All-seeing eyes are painted on the pavilion in the Nepali tradition.
The thirteen parasol rings of the spire represent attainment of first the ten bodhisattva levels and then the three bodies of the buddha. The parasol itself represents the wind element, the all-accomplishing wisdom and the karma family. The moon, sun and jewel represent the space element, dharmadhatu wisdom and the buddha family of beings.