text:
\section{Introduction}

The use of symmetry methods to understand bound states has an ancient origin and has given many of its fruits in atomic and subatomic physics - The work of Marcos Moshinsky is an excellent example of this \cite{mosh}. Over the years, however, other approaches to understand more complicated systems have given up symmetry in both classical and quantum mechanics \cite{stoeckmann}, \cite{luna}. Here we present one of the simplest ways of destroying continuous symmetries with purely quantum-mechanical consequences: A corner. We will show that a sharply bent waveguide supports a finite number of bound states whose nature has no classical counterpart. Due to the geometry, the traditional methods of integrability cannot be used to obtain the corresponding solutions.

This type of systems has been studied in the past, either with mathematical tools \cite{goldstone, exner1, exner2, exner3} proving the existence of bound states or by employing numerical methods for computing the corresponding energies and eigenfunctions \cite{exner1, schult, carini}. The approach in our considerations will be completely analytical, albeit we use approximations to perform computations. We want to show two different approaches to the problem in order to exhibit an apparent complexity of the description given in other studies, mostly numerical \cite{exnerl}. First we shall deal with a convenient basis of states which shows the binding as a consequence of the presence of corners: A sharp obstacle diffracts waves of an arbitrary wavelength. Some limitations of this method, such as the coupling of an infinite number of modes, will be pointed out. Finally, the use of conformal coordinates will be introduced as a natural description to find all the bound states for arbitrary bending angles.

We would like to emphasize that interesting applications may come in the form of bent wave guides carved in crystals, as suggested in \cite{opn}, using methods connected to the evolution of Airy profiles - a recent example can be found in \cite{paper by WPS}. The landscape looks promising, considering the many applications of controlling, bending and focusing waves.

\section{The existence of bound states}

\subsection{The oblique modes and the effective potential}

We show how to obtain a one dimensional Schroedinger equation with an effective potential along the longitudinal direction of the guide.
One of the challenges for describing our system is to find a suitable set of states. A natural way to introduce box eigenmodes in our system is by using oblique modes, as shown in the figure.

\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.4]{vguide2}
\end{center}
\label{fig1}
\caption{Description of our system. The distance between corners is denoted by $l$. The slope $a$ is related to the bending angle $\alpha$. We show an oblique mode with $n=2$}
\end{figure}

Let us take units such that $\hbar^2/2m=1$ with $m$ the mass of the particle and write the stationary Schroedinger equation in the form

\begin{eqnarray}
\nabla^2_{x,y} \psi + E \psi = 0
\label{1}
\end{eqnarray}
For the configuration in figure \ref{fig1} with Dirichlet conditions, we express the solution in the oblique basis as

\begin{eqnarray}
\psi(x,y) = \sum_{n=1}^{\infty} \psi_n(x) \sin \left( \frac{n \pi}{l}(y-a|x|+l) \right)
\label{2}
\end{eqnarray}
and $\psi(x,y)\neq 0$ for $a|x|-l<y<a|x|$. The matrix elements of the Schroedinger operator in this basis can be obtained by using the expansion (\ref{1}) and integrating over the $y$ variable. Let us denote the vector wave function by $\mbox{\boldmath$\Psi$\unboldmath}(x)$. Using the following definition for matrix elements

\begin{eqnarray}
\mathbf{N}_{nm} = \delta_{nm} n, \qquad \mathbf{M}_{mn} = (1-(-)^{m+n}) \frac{mn}{m^2-n^2}
\label{7}
\end{eqnarray}
and employing a gauge transformation given by

\begin{eqnarray}
\tilde{\mbox{\boldmath$\Psi$\unboldmath}}(x) = \exp \left( -\frac{2a|x|}{l} \mathbf{M} \right) \mbox{\boldmath$\Psi$\unboldmath}(x) \\
\tilde{\mathbf{N}}(x) = \exp \left( -\frac{2a|x|}{l} \mathbf{M} \right) \mathbf{N} \exp \left( \frac{2a|x|}{l} \mathbf{M} \right)
\label{8}
\end{eqnarray}
we reduce the Schroedinger equation to

\begin{eqnarray}
\left[\frac{d^2}{dx^2} - \left( \frac{\pi}{l} \right)^2 (a^2+1) \tilde{\mathbf{N}}(x)^2 - \left(\frac{2a}{l}\right)^2 \mathbf{M}^2 + E  \right] \tilde{\mbox{\boldmath$\Psi$\unboldmath}}(x)=0
\label{9}
\end{eqnarray}
where we see an effective potential of the form

\begin{eqnarray}
V_{eff} = \left( \frac{\pi}{l} \right)^2 (a^2+1) \tilde{\mathbf{N}}(x)^2 + \left(\frac{2a}{l}\right)^2 \mathbf{M}^2.
\label{10}
\end{eqnarray}
This matrix couples, in principle, all components of $\tilde{\mbox{\boldmath$\Psi$\unboldmath}}$. The origin of $\mathbf{M}$ and $\mathbf{N}^2$ is related to the momentum and the energy of a particle in a box. The $x$ dependence of the potential $V_{eff}$ comes completely from the non-commutability of $\mathbf{N}$ and $\mathbf{M}$ and is non-differentiable at the corner due to $|x|$. Another important feature is that a change of units  $x' = x/l, E'=E l^2 $ in equation (\ref{9}) leads to a scale-free equation, thus proving that the energy $E$ must scale as $l^{-2}$. Therefore, the presence of bound states in this problem is independent of the scale. This is in contrast with other bent systems with smooth boundaries \cite{exner1, exner2, exner3}.

\subsection{Solutions from corner-corrected WKB}

In previous works by Wheeler, Bestle and Schleich \cite{anti}, the WKB approximation was corrected in the presence of potentials with corners. As a result, the Schroedinger equation with a potential could be approximated by a WKB equation with a source term. By neglecting completely the effect of the potential except at the non-differentiable point, the WKB equation with a source could be interpreted as a free Schroedinger equation with a $\delta$ potential. In our case, this would cause a binding effect depending on the energy-dependent amplitude of the $\delta$. We shall apply this idea in order to cope with $V_{eff}$ obtained in the last section, with the additional feature that $V_{eff}$ is a matrix potential.

We start with the Schroedinger equation

\begin{eqnarray}
\frac{d^2 \mathbf{u}(x)}{dx^2} + \mbox{\boldmath$\lambda$\unboldmath}^{-2}(x) \mathbf{u}(x) = 0
\label{1.1}
\end{eqnarray}
where $\mbox{\boldmath$\lambda$\unboldmath} = \left(E-V_{eff} \right)^{-1/2}$ is now a matrix. The corner-corrected WKB formalism given in \cite{anti} can be easily generalized to matrix potentials. A WKB scattering equation (such as eq (13) in \cite{anti}) can be obtained for (\ref{1.1}). The resulting equation, in turn, can be treated by keeping first order terms in $a$ and considering a negligible potential except at $x=0$. This gives the isolated effect of the corner in the form

\begin{eqnarray}
\frac{d^2 \mathbf{u}(x)}{dx^2} + \left( E-\frac{\pi^2}{l^2} \mathbf{N}^2 \right)\mathbf{u}(x) = \mbox{$\cal V\,$}_{sc} \mathbf{u}(x) \nonumber \\
\mbox{$\cal V\,$}_{sc} \simeq -\frac{a \pi^2}{l^3} [\mathbf{N}^2,\mathbf{M}] \left( E- \frac{\pi^2}{l^2} \mathbf{N}^2 \right)^{-1} \delta(x).
\label{1.12}
\end{eqnarray}
The energies for bound states in this problem are obtained by i) imposing the appropriate boundary conditions $\mathbf{u}(x)=0$ at $x=\pm \infty$, ii) substituting the corresponding solutions in the condition for the jump in the derivative (due to $\delta$), namely

\begin{eqnarray}
\frac{d \mathbf{u}}{dx}\vert_{0+} - \frac{d \mathbf{u}}{dx}\vert_{0-} = -\frac{a \pi^2}{l^3} [\mathbf{N}^2,\mathbf{M}] \left( E-\frac{\pi^2}{l^2} \mathbf{N}^2 \right)^{-1} \mathbf{u}_0.
\label{1.13}
\end{eqnarray}
The solutions are found to be

\begin{eqnarray}
\mathbf{u}(x)= \mbox{$\cal N\,$} \left( \frac{\pi^2}{l^2}  \mathbf{N}^2 - E \right)^{-1/4} \exp \left( - |x| \sqrt{ \frac{\pi^2}{l^2} \mathbf{N}^2 - E} \right) \mathbf{u}_0
\label{1.14}
\end{eqnarray}
with $\mbox{$\cal N\,$}$ a normalization factor. The vector $\mathbf{u}_0$ is determined by the condition coming from the jump in the derivative (\ref{1.13}), namely

\begin{eqnarray}
\left[ \frac{a \pi^2}{l^3}  [\mathbf{N}^2,\mathbf{M}] + 2 \left( \frac{\pi^2}{l^2} \mathbf{N}^2 - E \right)^{3/2} \right] \left[ E - \frac{\pi^2}{l^2} \mathbf{N}^2 \right]^{-1} \mathbf{u}_0 = 0
\label{1.15}
\end{eqnarray}
This is a linear equation for $\mathbf{u}_0$ and the wave function can be obtained by choosing $\mathbf{u}_0$ as a member of the null space of the operator in (\ref{1.15}). The scaled energies $\tilde E = (l/\pi)^2 E $ are obtained from (\ref{1.15}) by using the determinant

\begin{eqnarray}
\det \left( \frac{a}{\pi} [\mathbf{N}^2,\mathbf{M}] + 2 \left[ \mathbf{N}^2 - \tilde E  \right]^{3/2} \right) = 0
\label{1.16}
\end{eqnarray}
The $2\times2$ case can be solved easily by inserting in (\ref{1.16}) the diagonal matrix $\left( \mathbf{N}^2-\tilde E \right)^{3/2}= \mbox{diag} \left( \left(1-\tilde E \right)^{3/2}, \left(4-\tilde E \right)^{3/2} \right) $. The resulting energy for the ground state is

\begin{eqnarray}
E_0 = \left(\frac{\pi}{l} \right)^2 \left( \frac{5}{2} - \frac{3}{2} \sqrt{1+\left( \frac{2}{3} \right)^2 \left( \frac{2a}{\pi} \right)^{4/3}} \right)
\label{1.18}
\end{eqnarray}
This energy is below the threshold $(\pi/l)^2$. The level $E_1$ can also be obtained; it is above $4(\pi/l)^2$. Therefore, in this $2\times2$ approximation, the only state which decays exponentially with the distance is the ground state. The wave function can be found straightforwardly once $u_0$ is obtained. We show two cases in figures \ref{fig2} \ref{fig2bis}.

The solutions thus obtained show that a bound state develops due to the corner alone, producing a $\delta$ potential and coupling the oblique modes - such coupling is essential, since a $1 \times 1$ approximation makes the interaction vanish. Yet, we may ask: where are the excited states below threshold due to an increasing value of $a$? In the following, we analyze this problem using conformal coordinates as a way to obtain a suitable basis. Such a basis cannot be obtained by means of perturbation theory in $a$ using the solutions of this section.

\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.6]{antitest11.pdf}
\includegraphics[scale=0.6]{antitest12}
\end{center}
\caption{Ground state wavefunction for $a=0.1$. A peak in the relevant region is visible. The decay with distance is exponential}
\label{fig2}
\end{figure}

\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.6]{antitest21}
\includegraphics[scale=0.55]{antitest22}
\end{center}
\caption{Ground state wavefunction for $a=0.2$. The peak becomes more pronounced as $a$ increases. For large values of $a$, the description is no longer valid.}
\label{fig2bis}
\end{figure}

\section{The conformal map}

Our aim now is to show that a non-trivial boundary value problem such as the bent waveguide can be mapped smoothly to a straight guide with an effective potential (equivalently, a problem with position-dependent mass). The simplest way to describe this process is by using complex variables. Denote $z=x+iy$, $\xi=u+iv$ and consider a transformation such that $\xi=F(z)$. We impose the following conditions on $F$:

1) The image of the bent waveguide is an infinite straight strip described by $0 < u < F(c)$.

2) The Laplacian $\nabla^2_{xy}$ is transformed into a quadratic form without cross-terms (The matrix $\mathbf{M}$ will not appear).

3) The transformation is smooth except at the points $O$ (upper corner) and $c$ (lower corner) in figure \ref{map2}.

The condition 1) allows the use of a complete set of functions $\lbrace \sin(nu/F(c))\rbrace$ for the Dirichlet problem and $\lbrace \cos((n+1/2)u/F(c))\rbrace$ for the Neumann problem, given that the contour lines meet the boundaries orthogonally. The condition 2) precludes the appearance of terms proportional to $\partial_u$: The integral $\int_{0}^{\pi F(c)} \sin(nu/F(c)) \partial_u \sin(mu/F(c))$ would produce unbounded matrix elements due to well-known properties of the momentum operator for a particle in a one-dimensional box. Requirement 3) ensures that any suspicious behavior of the transformation is due to the corners.

\begin{center}
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{conformalguide}
\end{center}
\caption{Contour lines of the coordinates $u,v$. We denote the lower corner by $c$ and place the upper corner at the origin $O$, see the text.}
\label{map2}
\end{figure}
\end{center}

\subsection{Explicit transformation}

The requirements described above are fulfilled by choosing $F$ analytic for all the points in the interior of the waveguide. Orthogonality is reached by the Cauchy-Riemann conditions. We introduce the operators

\begin{eqnarray}
\partial_{z} = \partial_x - i \partial_y, \qquad \partial_{z^*} = \partial_x + i \partial_y
\label{2.1}
\end{eqnarray}

\begin{eqnarray}
\partial_{\xi} = \partial_u - i \partial_v, \qquad \partial_{\xi^*} = \partial_u + i \partial_v
\label{2.2}
\end{eqnarray}
obtaining thus $\nabla^2_{xy}= \partial_{z}\partial_{z^*}$, $\nabla^2_{uv}= \partial_{\xi}\partial_{\xi^*}$. Using the analytic function $\xi=F(z)$, the Laplacian operator is transformed as $\partial_{\xi}\partial_{\xi^*} = |F'(z)|^{-2} \partial_{z}\partial_{z^*}$. Using the Cauchy-Riemann conditions we have $|F'(z)|^2 = \partial(u,v)/\partial(x,y)$, {\it i.e.\,} the Jacobian of the transformation as a function of $z$. Our boundary value problem with the Schroedinger equation becomes

\begin{eqnarray}
\left( \partial_{\xi}\partial_{\xi^*} + |\frac{dz}{d \xi}|^{2} E \right) \psi(u,v) = 0, \qquad u \in [0,F(c)], \quad v \in \bf{R}
\label{2.5}
\end{eqnarray}
We give an explicit example of our conformal map in the following. Consider the Schwarz-Christoffel (SC) transformation \cite{conformal} of the upper half-plane onto the semi-infinite strip. By using a composition of a SC map for a trigon with internal angles $\pi/2$ together with a SC map for a trigon with internal angles $ \alpha \pi$ and $(1-\alpha) \pi$, we arrive at the bijective transformation between one half of the bent waveguide and a semi-infinite straight strip. The expression results in

\begin{eqnarray}
\xi = F(z) = \int_{0}^{\sin^2(z/2)} dx (1-x)^{-\alpha} x^{-\alpha} = \int_{0}^{z} d\phi (\tan(\phi/2))^q \nonumber \\
= \beta \left(\sin^2(z/2), \frac{1-q}{2},\frac{1+q}{2} \right)
\label{2.9}
\end{eqnarray}
where we have used the definition of the incomplete beta function \cite{gradshteyn} and we have defined $q=1-2\alpha$ as the complementary angle of $2\alpha$ in $\pi$ radians. Therefore $0<q<1$. In passing we note that $F(c)=\sec (\pi q/2) $. The Jacobian is given by $|\frac{dz}{d\xi}|^2 = |\cot z/2|^{2q}$ and one can approximate it in terms of $\xi$ (or $u,v$) for some cases of interest:

\begin{eqnarray}
|\frac{dz}{d \xi}|^2=\cases{f(q)|\xi|^{-2q/(q+1)} & near the origin \cr g(q)|F(c)-\xi|^{2q/(1-q)} & near $F(c)$\cr 1+4q\cos u e^{-|v|}& away from the corners}
\label{2.14}
\end{eqnarray}
with $f(q)=((1+q)/2)^{-2q/(q+1)}$, $g(q)=(2(1-q))^{2q/(1-q)}$. It is important to note that as one moves away from the corner in the direction of the arms, the Jacobian (\ref{2.14}) becomes a hyperbolic cotangent of a real variable and approches to unity exponentially fast. Therefore, interactions vanish for points far from the corners. In connection with the existence of other maps, we stress that any other transformation constructed in this way will have the same behavior near the corners: The fractional power law of the map is completely determined by the angle of the guides and therefore by our physical system.

\subsection{Approximate solutions in conformal coordinates}

One can try many methods of solution for the transformed boundary value problem given above. Here we give an example based on the fact that our careful construction allows certain approximations. Let us reduce (\ref{2.5}) to a one-dimensional problem by defining $\epsilon_n \equiv -n^2/F(c)^2$ and

\begin{eqnarray}
V_{nm}(v) \equiv \< n | |\frac{dz}{d \xi}|^2 | m \> = \frac{2 F(c)}{\pi} \int_{0}^{\pi F(c)} du \sin \left( \frac{nu}{F(c)} \right) \sin \left( \frac{mu}{F(c)} \right) |\frac{dz}{d \xi}|^2
\label{2.6}
\end{eqnarray}
The set $\sqrt{2 F(c) / \pi} \sin(nu/F(c)) $ is obviously orthonormal. After multiplying (\ref{2.5}) by our complete set of functions and performing the corresponding integrals over $u$, we find that (\ref{2.5}) is equivalent to

\begin{eqnarray}
\sum_{m} \left(- \delta_{nm} \partial_v^2 - E V_{nm}(v) \right) \psi_m(v) = \epsilon_n \psi_n(v)
\label{2.8}
\end{eqnarray}
Near the origin, our potential (\ref{2.6}) becomes

\begin{eqnarray}
V_{nm}(v)= \frac{2 f(q)}{\pi F(c)} \int_{0}^{\pi F(c)} du
\sin \left( \frac{nu}{F(c)} \right) \sin \left( \frac{mu}{F(c)} \right){\left( u^2 + v^2 \right)^{q/(q+1)}}
\label{2.15}
\end{eqnarray}
It is justified to approximate the potential by its diagonal entries $V_n(v)$, since $V_{nm}(v)$ decreases as we move away from the diagonal for fixed $v$. This can be seen by noting that for $n \neq m$ the integrand can take negative values and oscillates rapidly as $n-m$ increases. We have

\begin{eqnarray}
\left(-\partial^2_v -E V_n(v) \right) \psi_n(v) = \epsilon_n \psi_n(v).
\label{2.16}
\end{eqnarray}
We include some plots of $V_n(v)$ and its derivative for different angles, figures \ref{pot}, \ref{slope}.

\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{potentials2}
\end{center}
\caption{A set of potentials for $n=1$, $q$ increases from $0$ to $1/2$ in $10$ steps. The transversal mode energy is shown at $-1$.}
\label{pot}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{slopes2}
\end{center}
\caption{The slope of the potentials as a function of $v$. Near the origin we see an abrupt change (a discontinuity appears in the limit $q=1$)}
\label{slope}
\end{figure}

The potentials shown in the figure have zero derivative at $v=0$ (except for $q=1$) but, apart from that, their behavior is that of a shallow potential well whose derivative suffers an abrupt change in a small region near the corner. In the plot showing the slopes, we see that there is almost a jump in $V'_n(v)$ around $v=0$. The formalism developed by Wheeler, Schleich and Bestle \cite{anti} in order to deal with 'kinks' should be helpful here again. Although the only case in which the derivative is strictly discontinuous corresponds to $q=1$, the strong variation of $V_n'(v)$ occurs in a region which looks effectively like a corner for long wavelengths (we expect this approximation to work at low energies). As the simplest approximation, let us consider the corner-corrected WKB formula for quantization of bound states, namely

\begin{eqnarray}
S = \sqrt{2} \int_{0}^{v_0} dv \sqrt{\epsilon_n+E V_n(v)}, \qquad \mbox{Classical orbit action}
\label{2.17}
\end{eqnarray}

\begin{eqnarray}
4S=2(k+{\textstyle{1\over2}})\pi - 4 \arctan (\frac{\Delta}{4}), \qquad \mbox{Phase shift}
\label{2.18}
\end{eqnarray}

\begin{eqnarray}
\Delta = E \left( V'_n(0+) - V'_n(0-) \right) \left( 2(\epsilon_n+E V_n(0)) \right)^{-3/2}, \qquad \mbox{Jump in the slope}
\label{2.19}
\end{eqnarray}

Remarkably, one can obtain good results by neglecting the classical orbit $S=0$ and keeping the corner effect alone. Using (\ref{2.17}), the approximation implies

\begin{eqnarray}
\frac{\Delta}{4} = \tan \left( \frac{2k+1}{4} \pi \right) = 1, \qquad k \quad \rm{even}
\label{2.20}
\end{eqnarray}
finding thereby that all even states obey this relation, but the excited states are not supported by such a weak potential. The excitations come from the solutions of (\ref{2.20}) by varying $n$ and not $k$. This results in a transversal node structure, instead of longitudinal. With these considerations, the solution to our problem (\ref{2.20}) requires $V_n(0)$ and $V_n'(0)$ as the only input. Such quantities can be estimated by means of (\ref{2.15}) for $m=n$. As mentioned before, we replace the jump in the derivative
$ V'_n(0+) - V'_n(0-)$ by twice the maximum of $V'_n(v)$ around the origin (see the figure). We denote such quantity as $V'_{max}$. Squaring (\ref{2.20}) we obtain a cubic relation for $E$ :

\begin{eqnarray}
\left( n^2 \cos^2 (\pi q/2) - E V_n(0) \right)^3 = \frac{1}{32} E^2 (V'_{max})^2
\label{2.22}
\end{eqnarray}
One can estimate $V_n(0)$ and $V'_{max}$ as

\begin{eqnarray}
V_n(0) \simeq \left( \cos (\pi q/2) \right)^{2q/(q+1)}, \quad V'_{max} \simeq \frac{1}{4} q \left(\cos (\pi q/2) \right)^{(2q+1)/(q+1)}.
\label{2.24}
\end{eqnarray}
With this, the energies in (\ref{2.22}) can be expressed in terms of $n$ and $q$ using Cardano's formula. We present the results graphically in figure \ref{final}.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.5]{phaseshiftenergies}
\end{center}
\caption{Energies as a function of the angle. There is a bound state for all $q>0$ as a consequence of the corner. An unlimited number of bound states appear as $q$ increases.}
\label{final}
\end{figure}

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.45]{conformaldensity2}
\end{center}
\caption{Contourplot of the probability density for $q=0.5$ and $n=1$ (ground state energy $E \simeq 0.56$). The peak of the distribution is located near the upper corner, as expected. This is achieved with only one transversal mode in conformal coordinates}
\label{density}
\end{figure}

The agreement of the estimated solutions with previous numerical results \cite{carini, dietz} is qualitatively good, considering that the action has been completely neglected in favor of the phase shift. Also, the off-diagonal elements of the matrix potential (\ref{2.15}) have been ignored. The trend of the curves as functions of the bending angle is comparable with the numerical results obtained in \cite{carini} for several modes: the energies decrease and more states appear below threshold as $q \mapsto 1$. The node structure of our wavefunctions is also compatible with the solutions obtained in \cite{carini} both numerically and experimentally. Remarkably, the fractional power behavior of the Jacobian alone is responsible for the structure of the spectrum of this system. We include a plot of the resulting wave density for the ground state of an L-shaped guide.

In what touches the corrections, we expect the following: The action appears to be important as $q$ increases (deeper wells). Therefore (\ref{2.22}) receives corrections which increase the energy. As a consequence, we expect a shift of the critical angles towards the value $q=1$. Precisely at this value of $q$ we also expect a non-zero lower bound for the energy $E_{min} \sim 1/4$, as can be seen by identifying such an extreme case with a guide of double thickness.

\section{Discussion}

In the first part we have shown that the existence of bound states could be understood as a corner effect and the coupling of transversal modes. We could learn that diffraction was essential to the final result, but the coupling of the modes gave an extra complication. The second part, however, shows that an appropriate set of coordinates may keep the effect of the corner without the coupling of such modes. This makes contact with recent dicussions \cite{teufel} in which perturbation theory seems doomed in the presence of unbounded curvature - our example fits perfectly in such class of problems. One can follow either approach to solve the problem, but its apparent complexity - e.g. the many modes used in \cite{exnerl} for the L-shape - dissapears using the second method. At the end we can conclude that a simple way of breaking a continuous symmetry such as a corner still allows for a simple description as long as the important information is used: diffraction and conformality.

The next task is to find the correction to the energy levels due to the classical orbits entering through the action. But we can say that for now our goal has been reached. As a final comment, the problem of bound states in multiple junctions -e.g. X-shaped and $*$-shaped geometries - lends itself for this kind of treatment by using Neumann boundary conditions. Due to the properties of our conformal coordinate system, this can be achieved by replacing $\sin(nu/F(c))$ by $\cos((n+{\textstyle{1\over2}})u/F(c))$ in our discussion.

\section*{Acknowledgments}
E. Sadurn\'i is grateful to the organizers of the Symposium {\it Symmetries in Nature\ } for their kind hospitality.


text:
\section{Introduction}
Whenever an image containing a human face is captured and analyzed, determining head pose often represents a crucial and challenging task for a wide number of applications such as biometric recognition or video surveillance, just to name the most obvious. This particular kind of intra-class variation can significantly affect recognition accuracy as well as have an impact  on any application requiring the normalization of head rotation with respect to its three degrees of freedom. The generalization of this problem from single image analysis to video sequence analysis, further extend its relevance and reach, which is typically proportional to absolute head rotation values.
Not surprisingly, then, head pose estimation represents a  hot research topic that has been approached by a large number of methods, most of which are based on the use of convolutional neural networks (CNNs) and deep learning (DL) techniques, often providing high accuracy with the not negligible cost of a prior training stage requiring positive and negative examples as input stimula.
Aiming at rivalling the accuracy of CNNs and DL based approaches, yet without the need of previous training process, we present HP2IFS a novel Head Pose estimation method based on Partitioned Iterated Function Systems (PIFS) and fractal image coding. PIFS, typically used for "lossy" image compression applications, is exploited as a mean to encode auto-similarities within the face image. The resulting fractal parameters vector is therefore compared to an angular reference array previously built through the same fractal coding algorithm by means of Hamming distance metric. Experiments conducted on both the Kinect Head Pose Database (BIWI) and the Automated Facial Landmarks in the Wild dataset (AFLW2000), confirm the validity of the proposed approach and its high accuracy compared to the state-of-the-art. The rest of this paper is organized as follows. Section II. resumes relevant works related to the present method. Section III. recalls the basics of fractal image coding and PIFS, while Section IV. describes in detail the proposed approach. Finally, Section V. presents the results of the experiments conducted and Section VI. concludes drawing some directions for future research.

\begin{figure}[!ht]
\centering
  \includegraphics[width=.48\textwidth]{codec1.png}
  \caption{The transformations between the domain blocks (D) and the range blocks (R), on image pairs featuring similar angular values.}
  \label{fig:codec1}
\end{figure}

\section{Related Works}

As already outlined in the introduction, a plethora of head pose estimation methods has been proposed throughout the last decades according to literature surveys such as \cite{trivedi2009} and \cite{Czup2014}. The majority of contributions to this topic have been based on machine learning approaches through deep neural architectures and more eminently on CNNs which are particularly suited to image processing. First works exploited Gaussian processes \cite{rasmussen2004gaussian}, or partial least square (PLS) regression techniques \cite{abdi2003partial} possibly implemented on support vector machines (SVMs) \cite{smola2004tutorial}. More recently, Drouard et al. \cite{drouard2017robust} proposed an approach to regression aimed at robust head pose estimation in the presence of intra-class variations, by mapping feature vectors associated to bounding boxes of faces onto the space of pose-angles and boundig-box translations. In \cite{kumar2017kepler} face keypoints detection and pose prediction are achieved by learning efficient regressors through an original neural architecture named Heatmap CNN, capable of extracting both global and local features to improve prediction reliability. Aiming at simultaneous face detection, keypoints localization, pose estimation and gender recognition, Ranjan et al. \cite{ranjan2019hyperface} make full use of deep CNN to develop a synergetic approach to these tasks that result in a boost up of each single task performance.
A total 3D solution to face alignment even for large poses (greater than 45° and up to 90°) is proposed by Zhu et al. \cite{zhu2017face} fitting 3D Morphable Models to 2D input face image by means of a cascaded CNN.
With regard to assessing the limits of deep NN for face alignment problem on existing 2D and 3D public datasets, the authors \cite{bulat2017far} develop a new public 3D facial landmark dataset named LS3D-W unifying all existing datasets and study the performance of both 2D and 3D pose estimation networks in this new context.
In the quest for a method not relying on facial keypoints, Ruiz et al. \cite{finegrained} propose to train a multi-loss CNN on the 300W-LP sinthetic dataset to achieve yaw/pitch/roll prediction through joint binned pose classification/regression from intensity images.
On a similar line of research, the authors of \cite{yang2019fsa} introduce a method without landmarks named FSA-Net, which relies on a soft stagewise regression scheme combining feature maps from different layers/stages.
In \cite{hsu2018quatnet}, a method combining L2 regression loss combined with an ordinal regression loss is used to train a CNN for highly accurate face pose estimation by addressing the ambiguity related to Euler angle representation through quaternions.
Wang et al. \cite{wang2019deep} exploits DL in a coarse-to-fine strategy based on two sub-networks trained jointly, where the coarse stage classifies the input face into four bins and the fine regression stage delivers pose angles.
Among the few works not based on machine learning methods, in \cite{Ricciardi_pose} a quad-tree based vector representing facial landmarks locations is compared to a pre-computed set of 2223 angular variations to estimate head pose to determine the closest approximation to the actual head pose.

The proposed HP\textsuperscript{2}IFS method, besides not requiring any prior training, not even requires any landmark detection, since it only relies on intrinsic image auto-similarities.

\section{Fractal Image Coding Basics}
\label{sec_frac}
The fundamental principle of fractal coding consists of the representation of an image by a contractive transform of which the fixed point is close to that image. This fixed point is named as \textit{fractal} \cite{fisher1995}.  Banach's fixed point theorem guarantees that, within a complete metric space, the fixed point of such a transform may be recovered by iterated application thereof to an arbitrary initial element of that space \cite{distasi2005range}.
Fractal coding techniques are based on the collage theorem, which  provides a bound on the distance between the image to be encoded and the fixed point of a transform, in terms of the distance between the transform of the image and the image itself. In the early 1980s, Barnsley \cite{barnsley1993fractal} applied his knowledge of fractal and mathematics to image compression, explaining that this transform was composed of the union of a number of affine mappings on the entire image - an iterated function system (\textit{IFS}) \cite{fisher1995}.
Fractal compression became a practical reality with the introduction by Jacquin of the \textit{partitioned}-IFS (\textit{PIFS}), which differs from an IFS in that each of the individual mappings operates on a subset of the image, rather than on the entire image \cite{fisher1995}.

\noindent The basic principle is that an image can be reconstructed by using the self-similarities in the image itself. When encoding an image, the algorithm partitions the original image into non-overlapping domain regions; after this, a new partition into smaller blocks (range blocks) is applied. For every range block, the best matching domain block is searched among all domain blocks by performing a set of transformations on the blocks. The compression is obtained by storing only the descriptions of these transformations.

\noindent A number of algorithms have been proposed by various authors to increase the efficiency of this method; the differences between the majority of existing fractal coding schemes may be classified into the following categories \cite{distasi2005range}:
\begin{itemize}
    \item partition imposed on the image support by the range blocks;
\end{itemize}
\begin{itemize}
    \item composition of the pool of domain blocks;
\end{itemize}
\begin{itemize}
    \item class of transforms applied to the domain blocks;
\end{itemize}
\begin{itemize}
    \item type of search used in locating suitable domain blocks;
\end{itemize}
\begin{itemize}
    \item representation and quantization of the transform parameters.
\end{itemize}

\subsection{Fractal Image Coding Implementation}
\noindent The fractal image encoding algorithm \cite{2015hybrid} works, in principle, as follows:\\

\noindent \textit{Domain Blocks}. The original input image is partitioned into non-overlapping $R$ blocks of size $N$ $\times$ $N$ to form $R$ block set \{$R$\}.\\

\noindent \textit{Range Blocks}. The input image is also divided into overlapped $D$ blocks to form D block set  \{$D$\}, with size $2N$ $\times$ $2N$.\\

\noindent \textit{Transformation}. The contractive transformation $w$ is a combination of a geometrical transformation and luminance transformation. The transformed version of the domain block can be rotated, mirrored, contrast scaled or translated, so the transformation can be shown as an affine transformation.\\

\noindent All $D$ blocks are contract to the size $N$ $\times$ $N$ (the same size of $R$ blocks) by downsampling or averaging the intensities of its four neighboring pixels.

\noindent The main computational step in fractal image coding is the mapping of domains to range blocks; contracted $D$ block is extended through $8$ isometric transformations (including identical transformation; rotations by $\ang{90}$, $\ang{180}$ and $\ang{270}$  about the block center; symmetric reflections about central vertical axis, horizontal central axis, leading diagonal and secondary diagonal). The extended domain pool that constructs a codebook is defined as \{$\Tilde{D}$\}. In order to find the best matching block for each $R$ block, contraction mapping transformation $W_i$ is defined as:

\begin{equation}
W_i =
    \begin{bmatrix}
    x\\
    y\\
    z\\
\end{bmatrix}
+
    \begin{bmatrix}
    a_i&b_i&0\\
    c_i&d_i&0\\
    0&0&s_i\\
\end{bmatrix}
    \begin{bmatrix}
    x\\
    y\\
    z\\
\end{bmatrix}
+
    \begin{bmatrix}
    e_i\\
    f_i\\
    o_i\\
\end{bmatrix}
\end{equation}

\noindent where $x$ and $y$ represent spatial coordinates,  $z$ indicates pixel value; $a_i$, $b_i$, $c_i$ and $d_i$ define one of the $8$ isometric transformations; $s_i$ represents brightness adjustment factor with its absolute value smaller than $1$ and $o_i$ notes brightness offset factor.

\noindent This operation allows to find the best matching block from the extended domain pool \{$\Tilde{D}$\} for each $R$ block, satisfying the minimum distortion error $E(R,\Tilde{D})$ of Eq. (\ref{eq:2}).

\begin{equation} \label{eq:2}
  E(R,\Tilde{D}) = ||R-(s \cdot \Tilde{D} + o \cdot I)||
\end{equation}

\noindent After fractal parameters of all $R$ blocks are stored as the result of compression, the total fractal encoding process is completed. In Figure~\ref{fig:codec1} we can see the transformation of the block of the domain in the block of the range. Images shown are from the dataset AFLW2000, described in Section V., and to which we have applied a mask, as described in Section~\ref{sec_met}. The two images are labeled with the same head pose, $\ang{5}$ Pitch, $\ang{30}$ Yaw and $\ang{-5}$ Roll. Due to the self-similarity induced by the fractal codec, we obtain that the same blocks of Domain will go in the same blocks of Range for both images within an acceptable margin of error Figure~\ref{fig:codec_details}. We use this propriety to build our method described in the following section.

\section{Method Description}
\label{sec_met}
HP\textsuperscript{2}IFS involves the use of fractal image coding algorithm to estimate face's pose and is structured in three main steps resumed below and depicted in Figure~\ref{fig:met}:
\begin{itemize}
    \item Step 1, \textit{Face detection and extraction};
\end{itemize}
\begin{itemize}
    \item Step 2, \textit{Fractal image coding algorithm} to obtain a matrix representing the fractal codes;
\end{itemize}
\begin{itemize}
    \item Step 3, \textit{Pose estimation}, transforming the fractal parameters in an array and comparing it to the angular array references obtaining in the same way (fractal codes) through Hamming distance metric.
\end{itemize}

\begin{figure}[!ht]
\centering
  \includegraphics[width=.5\textwidth]{codec_details.png}
  \caption{A detail of the fractal coding process, in particular the rotation and the lighting variation leading to the range located in row 8 and column 2.}
  \label{fig:codec_details}
\end{figure}

More in detail, given an input image, a face localization algorithm is applied to detect key facial structures on the region of interest; a face alignment is made based on a group of detected facial landmarks, to realize the accurate detection of the face-ROIs. This step produce a masked image where non-face pixels are set to black.
Afterwards, the Viola-Jones algorithm \cite{violajones} is used to detect the presence and the location of the face in the image, resulting in a square region containing the face. The method resizes always the original image in 256 $\times$ 256 pixels. The fractal codes for all images are calculated using the original fractal image coding algorithm described in Section~\ref{sec_frac}. These fractal parameters contain information about luminance and geometrical position for the range blocks, the domain block corresponding to each range blocks and the geometrical transformation like rotation and resizing to match the domain block with the range block.
The algorithm transforms the parameters resulting from the fractal compression into an encoding-array, creating a model matrix. Finally, a comparison with the model is performed through the Hamming distance metric, which is particularly suited to compare two data strings having the same length and is defined as the number of bit positions in which the two bits are different according to:

\begin{equation} \label{eq:ham}
  d(s,t)=\sum_{i=1}^n \delta(s_i,t_i)
\end{equation}

\begin{figure*}
\centering
  \includegraphics[width=.9\textwidth]{flow.png}
  \caption{Workflow of the proposed method.}
  \label{fig:met}
\end{figure*}

where $s$ and $t$ are the strings to compare having lenght $n$ and $\delta(s_i,t_i)$ is the following function

\begin{equation} \label{eq:ham2}
  \delta(s_i,t_i)=
  \begin{cases} 1, & \mbox{if }s_i\neq t_i \\
  0 & \mbox{if }s_i=t_i
\end{cases}
\end{equation}

This metric is notoriously simple to implement and fast to compute because its time complexity linearly depends on strings length. The minimum value obtained through the Hamming Distance between the model arrays and the input array give us the corresponding most similar head pose and its corresponding pitch, yaw and roll values.

\section{Experimental results}
As anticipated in the introduction, We relied on BIWI \cite{Biwi2011}  and AFLW2000 \cite{AFLW} datasets to perform the experiments. These two databases are completely different in terms of cooperativeness of the subjects and, as typically found in literature, we observed better results on BIWI in which subjects are cooperative. In order to test our method, as descibed in section~\ref{sec_met}, we manipulate the two datasets in two different ways to obtain the model set and the test set. \\
We tested HP\textsuperscript{2}IFS using a well known performance index for pose estimation, the Mean Absolute Error (MAE). MAE measures the average magnitude of the errors in a set of predictions and can be defined as follow:

\begin{equation} \label{eq:MAE}
  MAE=\frac{1}{n}\sum_{j=1}^{n} |y_j-\hat{y_j}|
\end{equation}
where $y_j$ is the ground truth, in our case the true angular value and $\hat{y_j}$ is the prediction, in our case the predicted angular value. We calculated the MAE for Pitch, Yaw and Roll separately and also an overall MAE of the error along the three axes.
In the following subsection we will describe results on BIWI and AFLW2000 in details.

\subsection{Results on BIWI}
The dataset BIWI offers a lot of image per identity. For this reason we decided to create our model using a one-left-out technique. We created ten different set of experimental data using in turn one individual as tester and the others as model. In particular, we have to emphasise, that every subject has a wide range of poses that cover our angular variation in terms of pitch, yaw and roll. The results about the MAE for each subset tested are shown in table~\ref{table:BIWIsub}. As can be seen from the table, there are also great variations in errors between an experimental subset and the other. This can be explained by the geometrical difference between the faces of the subjects. As an example, there are few subjects with a long face and the majority of the subjects have a round face. When the former is the tester, the results appear worst than on a subjects with a round face because there are less similar sample of this shape of face in the model. This is confirmed by the use of Fractal Coding (See Section~\ref{sec_frac}) that emphasize self-similarity between the sectors of an image.
The overall trend of the errors in terms of percentage of image tested can be seen in Figure~\ref{fig:ResBiwi}. We can observe a similar trend anticipated by the numerical results. In terms of Pitch, about 35\% of images have no error, 77\% of images have an error equal or less than $\ang{5}$, 95\% have an error equal or less than $\ang{10}$ and almost all images has an error less than $\ang{15}$.
In terms of Yaw, results are more encouraging. About 55\% of images have no error, 90\% of images have an error equal or less than $\ang{5}$, 97\% have an error equal or less than $\ang{10}$ and almost all images has an error less than $1\ang{15}$.
Roll achieved the best results because about 71\% of images have no error, 97\% of images have an error equal or less than $\ang{5}$, and almost all images have an error equal or less than $\ang{10}$.
\begin{figure}[!ht]
\centering
  \includegraphics[width=.45\textwidth]{biwi.jpg}
  \caption{Errors on BIWI in terms of percentage of tested images.}
  \label{fig:ResBiwi}
\end{figure}

Furthermore, we can analyse the errors in terms of angular variations. It is logical to assume that when head pose is more extreme, it is more difficult to estimate the angle accurately. From Figure~\ref{fig:HistBiwi} we can confirm that Pitch is the axis more affected by this issue, however the larger angular variation is in Yaw. In particular we can observe that, in general, the error decrease together with the estimated angle and increase when the estimated angle increase in modulus, confirming our hypothesis. In Table~\ref{tab:biwi} we show the comparison of HP\textsuperscript{2}IFS to state-of-the-art methods on BIWI. All methods consider are based on CNN or DL in general, with the exception of QT-PYR. Our method based on fractal coding achieves results comparable with recent techniques though they are based on machine learning approaches.

\begin{figure}[!ht]
\centering
  \includegraphics[width=.5\textwidth]{MAE-biwi1.png}
  \caption{Errors on BIWI in terms of angular poses.}
  \label{fig:HistBiwi}
\end{figure}

\begin{table}[h!]
\begin{center}
\caption{Results on the subsets of BIWI}
\label{table:BIWIsub}
\begin{tabular}{ |c|c|c|c|c| }
\hline
Subset & Yaw & Pitch & Roll & overall MAE \\
\hline
1 & 5.85 & 10.97 & 3.55 & 6.79\\
2 & 4.27 & 5.84 & 2.27 & 4.12\\
3 & 4.13 & 5.45 & 11.84 & 7.14\\
4 & 2.83 & 5.1 & 1.68 & 3.2 \\
5 & 3.46 & 5.13 & 1.89 & 3.49\\
6 & 2.9 & 4.79 & 1.68 & 3.12\\
7 & 4.57 & 8 & 2.56 & 5.04\\
8 & 5.5 & 5.92 & 3 & 4.8\\
9 & 3.66 & 5.09 & 2.58 & 3.77 \\
10 & 3.33 & 6.02 & 1.96 & 3.77 \\
\textbf{mean} & \textbf{4.05} & \textbf{6.23} & \textbf{3.3} & \textbf{4.52} \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[ht!]
\centering
\caption{Comparative results with the state-of-the-art on BIWI}
\label{tab:biwi}
\begin{tabular}{ |c|c|c|c|c| }
\hline
Method & Yaw & Pitch & Roll & MAE \\
\hline
Multi-Loss ResNet50~\cite{finegrained} & 5.17 & 6.97 & 3.39 & 5.177 \\
GPR~\cite{rasmussen2004gaussian} & 7.72 & 9.64 & 6.01 & 7.79 \\
PLS \cite{abdi2003partial}  & 7.35 & 7.87 & 6.11 & 7.11 \\
SVR \cite{smola2004tutorial} & 6.98 & 7.77 & 5.14 & 6.63 \\
hGLLiM \cite{drouard2017robust}  & 6.06 & 7.65 & 5.62 & 6.44 \\
QT PYR \cite{Ricciardi_pose} & 5.41 & 12.80 & 6.33 & 8.18 \\
FSA-Net \cite{yang2019fsa} & 4.27 & 4.96 & 2.76 & 3.996 \\
Coarse-to-Fine \cite{wang2019deep} & 4.76 & 5.48 & 4.29 & 4.84 \\
QuatNet \cite{hsu2018quatnet} & 4.01 & 5.49 & 2.93 & 4.14 \\
\hline
Proposed method & 4.05 & 6.23 & 3.30 & 4.52 \\
\hline

\end{tabular}
\end{table}

\subsection{Results on AFLW2000}
The AFWL2000 dataset is considered very challenging to head pose estimation. In order to obtain a subdivision necessary for HP\textsuperscript{2}IFS, we used about 80\% of the dataset images randomly selected to build the model and the remaining to perform the tests. As observed also in the state-of-the-art, results on AFLW2000 are worse than results on BIWI. However, we can observe a behaviour of the error along the three axis similar to that achieved on BIWI. As can be seen in figure~\ref{fig:ResAFLW}, the errors on Roll are lower than errors on Pitch and Yaw, but they show a similar behaviour for small errors.

 \begin{figure}[!ht]
\centering
  \includegraphics[width=.47\textwidth]{aflw2000.jpg}
  \caption{Errors on AFLW2000 in terms of percentage of tested images.}
  \label{fig:ResAFLW}
\end{figure}

In particular, for both Pitch, Yaw and Roll, between 25\% and 30\% of images have no error. Then, the results began diversifying along the three axes. With regard to Pitch, 60\% of images have an error equal or less than $\ang{5}$, 81\% have an error equal or less than $\ang{10}$, 92\% have an error equal or less than $\ang{15}$, 96\% have an error equal or less than $\ang{20}$ and almost all images has an error less than $\ang{25}$. In terms of Yaw, 66\% of images have an error equal or less than $\ang{5}$, 87\% have an error equal or less than $\ang{10}$, 95\% have an error equal or less than $\ang{15}$, 97\% have an error equal or less than $\ang{20}$ and almost all images has an error less than $\ang{25}$.
Finally, Roll, that shown also in this case the better results, 72\% of images have an error equal or less than $\ang{5}$, 92\% have an error equal or less than $\ang{10}$, 98\% have an error equal or less than $\ang{15}$, and almost all images has an error less than $\ang{20}$.

\begin{figure}[!ht]
\centering
  \includegraphics[width=.47\textwidth]{MAE-AFLW2000.png}
  \caption{Errors on AFLW2000 in terms of angular poses.}
  \label{fig:HistAFLW}
\end{figure}

Also in this case, we can observe the behaviour of the error compared to the angular variations, in Figure~\ref{fig:HistAFLW}. The intervals of variation in degree for the ground truth depends on the available images in AFLW2000. Even if with different kind of errors, the average behaviour along the three axis is the same. The error increase together with the angle in modulus, just like we have observed in the results of BIWI. In Table~\ref{tab:aflw2000} the results of HP\textsuperscript{2}IFS on AFLW2000, compared to state-of-the-art methods, are shown. Even though the performance is lower then on BIWI, our results are comparable with other methods, encouraging us to consider the current implementation of fractal coding a promising starting point for further improvements of HP\textsuperscript{2}IFS also on competitive datasets as AFLW2000.

\begin{table}[ht!]
\centering
\caption{Comparative results with the state-of-the-art on AFLW2000}
\label{tab:aflw2000}
\begin{tabular}{|l|l|l|l|l|}
\hline
Method & Yaw & Pitch & Roll & MAE \\
\hline
Multi-Loss ResNet50~\cite{finegrained} & 6.470 & 6.559 & 5.436 & 6.155 \\
Hyperface \cite{ranjan2019hyperface} & 7.61 & 6.13 & 3.92 & 5.89 \\
KEPLER \cite{kumar2017kepler} & 6.45 & 5.85 & 8.75 & 7.01 \\
3DDFA \cite{zhu2017face}  & 5.400 & 8.530 & 8.250 & 7.393 \\
FAN \cite{bulat2017far} & 6.358 & 12.277 & 8.714 & 9.116 \\
QT PYR \cite{Ricciardi_pose} & 7.6 & 7.6 & 7.17 & 7.45 \\
QuatNet \cite{hsu2018quatnet}  & 3.973 & 5.615 & 3.92 & 4.503 \\ \hline
Proposed Method & 6.28 & 7.46 & 5.53 & 6.42 \\ \hline
\end{tabular}
\end{table}

\section{Conclusions}
We presented HP\textsuperscript{2}IFS, a novel method for head pose estimation based on partitioned iterated function systems to extract a fractal image code from input face to be matched against a reference array through Hamming distance. The proposed approach is explicitly is aimed at achieving accurate angular values for yaw, pitch and roll axis without requiring any neural network architecture and the related training overhead. According to the results of experiments carried out on both BIWI and AFLW2000 reference datasets, the reported pose estimation error is comparable to that of state-of-the-art methods based on machine learning and often inferior to that reported by the best performing non-training based methods.We are currently considering new non-linear transformations capable of enhancing the concept of self-similarity, as well as designing an ad-hoc metric for the distance between IFSs.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

Fix a positive integer $a$, and  write $\mathrm{ord}_n(a)$  for the order of $a$ in the multiplicative group of invertible residue classes modulo $n$ when $(a,n)=1$. Let $$G(x)= \sum_{n \leq x, (n,a)=1} \frac{\phi(n)}{\mathrm{ord}_n(a)}.$$ We show that for any $\alpha <3$, we have $G(x) = O(x^2/ \log^\alpha x)$.  The motivation for investigating these sums stems from two distinct problems: the Artin primitive root conjecture and the problem of counting Artin representations.

Artin conjectured that for any given a rational integer $a$, $a \neq 1$ and $a$ not a perfect square,  the set rational primes $p$ such that $a$ is a primitive root mod $p$ has positive density and he conjectured a formula for that density. Major work on Artin's conjecture is due to Hooley \cite{Hooley}, Murty and Gupta\cite{Murty}, and Heath-Brown\cite{Heath-Brown}.  Since then, work has been done generalizing the Artin conjecture in a variety of directions.
Our result concerning $G(x)$ represents one such direction. Instead of confining ourselves to prime moduli, we consider also composite moduli and instead of considering moduli for which $a$ is of maximal order, we consider the average order of the index of the cyclic group generated by the residue class of $a$.

Our second motivating problem, the problem of counting Artin representations, is actually connected not to $G(x)$ itself but rather to a closely related sum over number fields. Let $K$ be a number field with ring of integers $O_K$. Let $U_K$ be the set of units of $O_K$, and for any ideal $I$ let $U_K(I)$ be the subgroup of $U_K$ formed by elements which are 1 (mod I). Let $U_K(I)^+$ be the subgroup of $U_K(I)$ formed by elements which are positive in all real embeddings of $K$.  Let $$P_K(x)= \sum_{\textbf{N}I \leq x} \frac{\phi(I)}{[U_K:U_K(I)]}$$ with the sum over non-zero ideals of $O_K$. Assume that the unit group of $O_K$ has positive rank. Then for any $\alpha <3$, we have $P_K(x) = O(x^2/ \log^\alpha x)$. $P_K$ represents an analog of $G$ for number fields.

Let us now explain the connection between these sums and the problem of estimating the number of Artin representations of fixed dimension, fixed base field, and  conductor of bounded norm. We focus on the case of dimension 1, so one is essentially counting primitive ray class characters. Let $K$ be a number field and let $\delta_{K,1}(x)$ count the number of 1-dimensional Artin representations up to isomorphism with norm of the conductor at most $x$.  Rohrlich\cite{Rohrlich} has noted the elementary estimate $\delta_{K,1}(x) = O(x^2)$, and if $K$  is the rationals or a quadratic imaginary field, then this is the correct order of growth and one can in fact produce an asymptotic formula, with the constant depending on the field. The method of proof is to note that one has
$$\delta_{K,1}(x) \leq \sum_{\textbf{N} I \leq x}{h_K}^{nar}(I)$$ where ${h_K}^{\mathrm{nar}}(I)$ is the order of the narrow ray class group of $K$ with modulus $I$. In fact, one has $$\delta_{K,1}(x) = \sum_{\textbf{N} I \leq x}\sum_{Q|I}\mu(\frac{I}{Q}){h_K}^{nar}(Q)$$ where $\mu$ is the generalization of the mobius function to ideals.

One has \cite{Lang2} $${h_K}^{\mathrm{nar}}(I)= \frac{2^{r_1}h_K\phi(I)}{[U_K:{U_K}^+(I)]}.$$  Note that ${[U_K:{U_K}^+(I)]}$ and ${[U_K:{U_K}(I)]}$ differ by at most a power of 2, the maximum exponent of which is bounded in terms of the rank of the unit group of the field. Thus, estimating $P_K(x)$ gives us  information about the number of Artin representations, since the other parts in the formula for the order of the  narrow class group are all dependent only on $K$.  Prior to this work, lower bounds for both $P(x)$ and $G(x)$ have been obtained by  Ambrose\cite{Ambrose}, who also conjectured that for both functions, the correct growth order would be $x^{2+o(1)}$. The estimates for $G(x)$ and $P_K(x)$ are similar, as one can think of $G(x)$ as the analog to $P_K(x)$ in the $S$-integers with $S$ equal to the set of prime divisors of $a$, but the proofs are presented separately; it is likely that a framework can be constructed which subsumes both results into a single proof, but as of yet, attempts at such an approach lead to technical difficulties.

\section{Over the rational integers}

\begin{lemma} If $2 \leq y \leq \frac{x}{2}$, then $\frac{x^x}{(x-y)^{x-y}} < e^{2y}x^y$.
\label{lemma1}
\end{lemma}
\begin{proof}
Consider
$$\log \frac{x^x}{(x-y)^{x-y}} = x\log x -  (x-y)\log(x-y)=x\left(\log x - (1-\frac{y}{x})\log(x-y) \right).$$ We have $$x\left(\log x - (1-\frac{y}{x})\log(x-y) \right) < x\left(\log\frac{x}{x-y} + \frac{y \log x}{x}\right).$$
Set $u=\frac{x}{x-y}-1=\frac{y}{x-y}$. Since $y \leq x/2$ and $\log(1+u) \leq u$ we have $\log(\frac{x}{x-y}) \leq \frac{2y}{x}$.
Thus we have $\log \frac{x^x}{(x-y)^{x-y}} \leq x\left(\frac{2y}{x} + \frac{y \log x}{x} \right)=2 y + y\log x$, so exponentiating
now gives the desired result.

\end{proof}

\begin{lemma} If $1 \leq j \leq \frac{k-2}{3}$, then ${k \choose j} \leq \frac{{k \choose j+1}}{2}$.
\label{lemma2}
\end{lemma}
\begin{proof} Note that ${k \choose j}/{k \choose j+1}= (j+1)/(k-j)$.
We have $\frac{j+1}{k-j} \leq \frac{1}{2}$ when $j \leq \frac{k-2}{3}$.
\end{proof}

\begin{lemma} If $2 \leq m \leq \frac{k-2}{3}$ then $\sum_{j=1}^m {k \choose j} \leq \left(\frac{e^2 k}{m}\right)^m$.
\label{HRlemma}
\end{lemma}
\begin{proof} We will first estimate ${k \choose m}$ and then use Lemma \ref{lemma2}  to bound $\sum_{j=1}^m {k \choose j}$.
We require the following version of Stirling's formula, valid for $n \geq 2$,:
$$\sqrt{2\pi n}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n+1}} < n! < \sqrt{2\pi n}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n}}$$ (See for example Lang's {\it Undergraduate Analysis\cite{Lang}.}) We have $$ \frac{k!}{m!(k-m)!} < \frac{\sqrt{2\pi k}\left(\frac{k}{e}\right)^k e^{\frac{1}{12k}} }{\sqrt{2\pi m}\left(\frac{m}{e}\right)^m \sqrt {2\pi (k-m)}\left(\frac{k-m}{e}\right)^{k-m}e^{\frac{1}{12m+1}} e^{\frac{1}{12(k-m)+1}} }.$$
Since $m < k$, we obtain:
$${k \choose m} < \frac{\sqrt{k}k^k}{\sqrt{m}m^m \sqrt{2\pi (k-m)}(k-m)^{k-m}}.$$ Applying Lemma \ref{lemma1} with $x=k$ and $y=m$ , we have $\frac{k^k}{(k-m)^{k-m}} < e^{2m}k^m$,
and thus
$${k \choose m} < \frac{e^{2m}k^m\sqrt{k}}{m^m\sqrt{m}\sqrt{2\pi (k-m)}}.$$ Since $m \leq \frac{k-2}{3}<\frac{k}{2}$, $k-m >\frac{k}{2}$ and so,
$${k \choose m} < \frac{e^{2m}k^m\sqrt{k}}{m^m\sqrt{m}\sqrt{\pi k }} = \frac{e^{2m}k^m}{\sqrt{m}\sqrt{\pi}m^m}.$$ By Lemma \ref{lemma2},
we have that $$\sum_{j=1}^m{k \choose j} \leq {k \choose m} + \frac{{k \choose m}}{2} + \frac{{k \choose m}}{4} \cdots = 2{k \choose m}< 2\frac{e^{2m}k^m}{\sqrt{m}\sqrt {\pi}m^m}$$
Since $m \geq 2$, $\frac{2}{\sqrt{\pi m}} < 1$ so we are done.
\end{proof}

\begin{lemma} For $\beta>0$, let $D_\beta(x)$ be the number of positive integers $n$ with $\omega(n) \geq (\log x)^\beta$. Then for any fixed $\beta$, we have
$D_\beta(x) = O(\frac{x (\log \log)^2}{(\log x)^{2\beta}})$, where the implied constant depends on $\beta$.
\label{Hardy1}
\end{lemma}
\begin{proof}
This follows from 22.11.6 (page 357) in Hardy and Wright\cite{Hardy}
\end{proof}

\begin{theorem} For any $\alpha <3$, we have $G(x) = O(\frac{x^2}{\log^\alpha x})$
\end{theorem}
\begin{proof} We will prove the equivalent result that  for any fixed $\alpha$ such that $1< \alpha< 3$, we have $G(x) = O(\frac{x^2 (\log \log x)^2}{(\log x)^\alpha})$.

Let $S(x)$ be the set of $n \leq x$ with  $(n,a)=1$. Define $S(x,p,q)$ to be the subset of $S(x)$ with $p \leq \mathrm{ord}_n(a) \leq q$. Set $t = (\log x)^\alpha$.  Note that if $x$ is sufficiently large we have that $t > 3\log_a x$.  Write $\ell(x)=3\log_a x$. Then we have:

\begin{equation}\label{Gbreakdown} G(x) \leq \mathrm{I}(x) + \mathrm{II}(x) + \mathrm{III}(x)  \end{equation}
where \begin{equation} \label{Idef1} \mathrm{I}(x)= \sum_{n \in S(x,1,\ell(x))} \frac{n}{\mathrm{ord}_n(a)}\end{equation} \begin{equation}\label{IIdef1} \mathrm{II}(x)= \sum_{n \in S(x,\ell(x),t)} \frac{n}{\mathrm{ord}_n(a)}\end{equation} and \begin{equation}\label{IIIdef1} \mathrm{III}(x)= \sum_{n \in S(x,t, \infty) } \frac{n}{\mathrm{ord}_n(a).}\end{equation}  We will need to estimate each of these sums separately.

The easiest two sums to estimate are $\mathrm{I}(x)$ and $\mathrm{III}(x)$. $\mathrm{I}(x)$ has at  most $\sum_{1 \leq r \leq \ell(x)} \tau(a^r-1)$ terms. We have
$$\sum_{\substack{1 \leq r \leq \ell(x) }} \tau(a^r-1) \leq \ell(x)\tau_m(x)$$ where $\tau_m(x)$ is the maximum of $\tau(a^r-1)$ with $r$ ranging
from $1$ to $\ell(x)$. Since $\tau(n)=O(n^{1/6})$, $\tau_m(x)=O(x^{1/2})$. Any term in $I(x)$ is bounded above by $x$, and so we conclude that
$I(x)= O(x^{1/2} \ell (x)x) = O(x^{3/2}\log x)$.

Next, we need to estimate $\mathrm{III}(x)$. In this case we use that it has at most $x$ terms and each term is at most $x/t$ and so $\mathrm{III}(x) \leq x^2/t$.

To estimate $\mathrm{II}(x)$ we need some preliminary remarks.

Given any $\epsilon >0$,  we have $\omega(n) < \frac{(1+\epsilon)\log n}{\log \log n}$ for all but finitely many $n$. For simplicity, we take $\epsilon=1$ and so for all but finitely many $n$ we have  $\omega(n) < \frac{2\log n}{\log \log n}$. Let $N$ be the set of $d$ such that
$(a,d)=1$ and $d|n$ for some $n$ satisfying $\omega(n) \geq \frac{2\log n}{\log \log n} $. Note that $N$ is finite.
Since $N$ is finite, $\sum_{n \in N, n \leq x} \frac{\phi(n)}{\mathrm{ord}_n(a)} = O(1)$.

Set $\beta = \frac{\alpha-1}{2}$. Set $H(x)$ to be the subset of elements of $S(x,\ell(x),t)$ and not in $N$ which have at most $(\log x)^\beta$ distinct prime factors,  and set \begin{equation}\label{II1def1} \mathrm{II}_1(x)= \sum_{n \in H(x)} \frac{n}{\mathrm{ord}_n(a)}.\end{equation} Similarly, let $J(x)$ be the elements with more than $(\log x)^\beta$ distinct prime factors and not in $N$
and set  \begin{equation}\label{II2def1} \mathrm{II}_2(x) =  \sum_{n \in J(x)} \frac{n}{\mathrm{ord}_n(a)}.\end{equation} Thus, $\mathrm{II}(x)= \mathrm{II}_1(x) + \mathrm{II}_2(x) +O(1)$.  So we need only bound $\mathrm{II}_1(x)$ and $\mathrm{II}_2(x)$.

To estimate $\mathrm{II}_1(x)$, we need to estimate $H(x)$. If $d \in H(x)$, then $d|a^r-1$ with $\ell(x) \leq r \leq t$. So we need to estimate how many
 divisors $a^r-1$ can have which are less than or equal to  $x$ and not in $N$.  Note that the number of prime factors of $a^r-1$ is bounded by  $$\omega(a^r-1) \leq \frac{2\log a^r}{\log \log a^r} \leq \frac{(3\log a)r}{\log r} \leq t.$$ The last inequality above is valid for sufficiently large values of $t$. Recall that $t$ grows with $x$.

 Thus,  each $a^r-1$ whose divisors contribute to $H(x)$ has at most $t$ distinct prime divisors.
However, each element in $H(x)$ has at $j$ distinct prime divisors, for some $j$ satisfying $1 \leq j \leq (\log x)^\beta$. So for any given $r$, there are at most at most $\sum_{j=1}^m {k \choose j}$ possible choices for the distinct prime divisors where $k$ is the largest integer less than or equal $t=(\log x)^\alpha$
and $m$ is the largest integer less than or equal to $(\log x)^\beta$. Any prime factor of such a divisor can
be raised to at most the $\log_2 x$ power, so the total number of divisors is bounded by  $(\sum_{j=1}^m {k \choose j})(\log_2 x)^{{(\log x)}^{\beta}}.$ There are at most $t$ possible values of $r$.  Thus,
$$|H(x)| \leq t\sum_{j=1}^m {k \choose j}(\log_2 x)^{(\log x)^\beta}.$$  Applying Lemma \ref{HRlemma} and using the values of $k$ and $m$ then gives $$ |H(x)|\leq t\left(\frac{e^2 t }{(\log x)^\beta }\right)^{(\log x)^\beta} (\log_2 x)^{(\log x)^\beta}.$$ Here we can replace the floor of $(\log x)^\beta$ since $\left(\frac{e^2 t }{(s}\right)^s$ is an increasing function in $s$ when $s$ is small compared to $t$.

 Thus we have $$\log |H(x)| \leq C(\log x)^\beta(\log \log x) + \log t + (\log \log_2 x)(\log x)^\beta  $$ for some constant $C$ and this is $O((\log x)^\beta \log \log x)$. Thus $|H(x)| = O(x^{\epsilon_0})$, and so $II_1(x) = O(x^{1+\epsilon_0})$ for any $\epsilon_0>0$.

To estimate  $\mathrm{II}_2(x)$ we apply Lemma \ref{Hardy1} so that the number of terms of $J$ is $O(\frac{x (\log \log x)^2}{(\log x)^{2\beta}})$ and thus conclude that $$\mathrm{II}_2(x) = O\left(\frac{x^2 (\log \log x)^2}{(\log x)^{2\beta +1}}\right)$$ since every
term in $\mathrm{II}_2(x)$ is at most $\frac{x}{\ell(x)}$.
\end{proof}

\section{Number fields}

Let $K$ be a number field and let $O_K$ be its ring of integers. Set $d= [K:Q]$. Define  $\omega(I)$ to be the number
of distinct prime ideal divisors of $I$, where $I$ is an ideal of $O_K$. Define  $j_K(x)= \sum_{\textbf{N}I \leq x} \omega(I)^2$.

\begin{lemma} $j_K(x)= O(x(\log \log x )^2)$.
\label{idealboundjk}
\end{lemma}
\begin{proof} We generalize the argument in Hardy and Wright (who treat the case $K=Q$). Note that $$\sum_{\substack{\textbf{N}I \leq x \\ \omega(I)=1}} \omega(I) = O(x).$$ But if $\omega(I) \geq 2$, then $$\omega(I)^2 - \omega(I) \geq \frac{\omega(I)^2}{2}.$$  Thus, we have

$$j_K(x) \leq 2\sum_{\textbf{N}I \leq x } \omega(I)(\omega(I)-1)  +O(x).$$ Thus, we need only show that
 $$\sum_{\textbf{N}I \leq x } \omega(I)(\omega(I)-1)= O(x(\log \log)^2) .$$ For a given ideal $I$, consider distinct prime ideals $P$ and $Q$ which divide $I$. There are $\omega(I)$ possible choices for $P$ and $\omega(I)-1$ possible choices for $Q$. Thus we have $$\sum_{\textbf{N}I \leq x } \omega(I)(\omega(I)-1) = \sum_{\textbf{N}I \leq x}\left(\sum_{PQ|I} 1\right) $$ where the sum is over distinct prime ideals. So we need only estimate $$\sum_{\textbf{N}I \leq x}\left(\sum_{PQ|I} 1\right).$$

Now, let $k$ be a constant such that the number of ideals of norm at most $x$ is bounded  by $kx$. Then,
$$\sum_{\textbf{N}I \leq x}\sum_{PQ|I} 1 \leq \sum_{\textbf{N}(PQ) \leq x} \frac{kx}{\textbf{N}(PQ)}$$ where the sum is over $P, Q$ prime. So if we can show that $$\sum_{\textbf{N}(pq) \leq x} \frac{x}{\textbf{N}(PQ)} = O(x (\log \log x)^2)$$ we will be done.

There are two contributions to this sum, terms where $\textbf{N} P$ and $\textbf{N} Q$ are both prime ideals of degree 1, versus terms where at least one prime is of degree greater than 1. Consider first the terms where at least one of $\textbf{N} P$ and $\textbf{N} Q$ is a power $\geq 2$ of a prime. Without loss of generality, consider the terms where $\textbf{N} P$ is a power $\geq 2$ of a prime. Since there are at most $d$ primes ideals of norm $p^a$ (where $p$ is a rational prime) and similarly at most $d$ primes of norm $q$ where $q$ is a rational prime, we then have: $$\sum_{\textbf{N}{PQ} \leq x, \textbf{N}P=p^a} \frac{x}{\textbf{N}(PQ)} \leq \sum_{\substack{a\geq 2 \\ p, q \leq x}} \frac{d^2 x }{p^a q}.$$ Here the sum on the left-hand of the inequality is over prime ideals $P$ of degree greater than 1, and  over prime ideals $Q$ of any degree.   We have $\sum_{a \geq 2}\frac{1}{p^a} \leq \frac{2}{p^2}$ and so
$$\sum_{\substack{a\geq 2 \\ p, q \leq x}} \frac{d^2 x }{p^a q} \leq \sum_{{q \leq x, p \leq x^{1/2}}} \frac{2d^2 x }{p^2 q} \leq \left(\sum \frac{2d^2}{p^2}\right) \sum_{q^i \leq x}{\frac{x}{q^i}} = O(x \log \log x.)$$

Thus we need to only examine $\sum_{\textbf{N} PQ \leq x} \frac{x}{\textbf{N} PQ}$ with the sum taken over ideals with prime norm.
This sum is bounded by $$\sum_{p,q \leq x}\frac{x}{pq} \leq  x \sum_{p,q \leq x}\frac{1}{pq} \leq x(\sum_{p \leq x} 1/p)^2 = O(x (\log \log x)^2).$$
\end{proof}

As an immediate corollary of the above we obtain:

\begin{lemma}
\label{omegacountbound2}
For $\beta>0$, let $D_\beta(x)$ be the number of ideals $I$ with $\omega(I) \geq (\log x)^\beta$, and $\textbf{N}I \leq x$. Then for any fixed $\beta$, we have
$D_\beta(x) = O(\frac{x (\log \log)^2}{(\log x)^{2\beta}})$, where the implied constant depends on $\beta$.
\end{lemma}

For a matrix $A$ with coefficients in $\mathbb{C}$, we will denote by $M(A)$  the maximum of the absolute value of the
entries of $A$.

\begin{lemma} Let $A$ be an $n \times n$ matrix with coefficients in $\mathbb{C}$. Then $|\det(A)| \leq  n! M(A)^n.$
\label{determinantbound}
\end{lemma}
\begin{proof} We have $$\det(A) = \sum_{\sigma \in S_n} sgn(\sigma) \prod_{i=1}^n A_{i,\sigma(i)}.$$ So
$$|\det(A)| \leq \sum_{\sigma \in S_n} |\prod_{i=1}^n A_{i,\sigma_i}| \leq \sum_{\sigma \in S_n} M(A)^n = n! M(A)^n.$$
\end{proof}

\begin{lemma}
\label{maximummatrixentry} Let $A$ and $B$ be $n \times n$ matrices with coefficients in $\mathbb{C}$. Then $M(AB) \leq nM(A)M(B)$. Thus, for any $m$, $M(A^m) \leq (nM(A))^m$.
\end{lemma}
\begin{proof}For any $i,j,k,l,$ $|A_{ij}B_{kl}| \leq M(A)M(B)$.  Since every entry in $AB$ arises from a sum of $n$ terms each of which is a product of an entry in $A$ and an entry $B$, the desired bound follows. The second part of the lemma follows from the first.

\end{proof}

Henceforth, we will assume that $O_K$ has a unit $a$ of infinite order (that is $K$ is not the rationals or a quadratic imaginary field).

\begin{lemma}For any $a$ in $O_K^{\times}$,  there exists a constant $C>1$ such that for all $k$, $$|N_{K/\mathbb{Q}}(a^k-1)| \leq C^k.$$
\label{fieldnormbound}
\end{lemma}
\begin{proof} Pick some Archimedean embedding of $K$ into $\mathbb{C}$ and let $\lambda_1, \lambda_2, \cdots \lambda_d$ be a basis for $K$ over $\mathbb{Q}$. Then for some $a_{ij}$ we have $$a\lambda_i=\sum_j a_{ij}\lambda_j.$$
 We have $N_{K/\mathbb{Q}}(a)= \det(A)$ where $A$ is the matrix with entries $a_{ij}$. Then, $$N_{K/\mathbb{Q}}(a^k)= \det(A^k).$$ By Lemma \ref{maximummatrixentry} $$M(A^k) \leq (dM(a))^k.$$ Let $B$ be the matrix with coefficients $b_{ij}$ defined by $$(a^k-1)\lambda_i =\sum_j b_{ij}\lambda_j.$$ So $N_{K/\mathbb{Q}}(a^k-1) = \det B$. Note that $B$ depends on $a$ and $k$.  By Lemma \ref{determinantbound}, we  have $$|N_{K/\mathbb{Q}}(a^k-1)| = |\det B| \leq d!M(B)^d.$$ Since $M(B) \leq M(A^k)+1$, we can apply Lemma \ref{maximummatrixentry} and Lemma \ref {determinantbound} so $$ |\det B| \leq d!(dM(A)^k+1)^d \leq (d!(1+dM(A)))^{dk}.$$ So we may take $C = (d!(1+dM(A)))^d$.

\end{proof}

 For any ideal $I$ of $O_K$ let $o_a(I)$ be the smallest positive integer such that $I|(a^{o_a}-1)$. So, $o_a$ is the order of $a$ in the group $(O_K/I)^{\times}$.

\begin{lemma}
 There is a constant $C> 1$ depending on $K$ and $a$ such that for any ideal $I$ of $O_K$,  we have $o_a \geq \log_{C} \textbf{N}I$.
 \label{idealcountlemma}
\end{lemma}
\begin{proof} Assume that $I|(a^{k}-1)$. So $\textbf{N}I \leq \textbf{N}(a^k-1) = |N_{K/\mathbb{Q}}(a^k-1)|$ since $(a^k-1)$ is a principal ideal. By Lemma \ref{fieldnormbound}, there is a constant $C>1$ depending only on $a$ such that $|N_{K/\mathbb{Q}}(a^k-1)| \leq C^k$ from which the result follows.

\end{proof}

\begin{corollary} Assume that the group of units of $O_K$ has positive rank. Then there is a constant $c>1$ such that
$[U_K:U_K(I)] \geq \log_{c} \textbf{N}I$.

\end{corollary}
\begin{proof} Apply Lemma \ref{idealcountlemma}, and note that if $u$ is a unit of infinite order, then $I|(u^{[U_K:U_K(I)]}-1)$.
\end{proof}

\begin{lemma} There exists a constant $k$ such that for all but finitely many ideals $I$, we have
$\omega(I) < k\frac{\log \textbf{N}I}{\log \log \textbf{N}I}$. Moreover, if we let $\tau(I)$ be the number of distinct ideal divisors of $I$ then
$\tau(I) = O((\textbf{N}I)^\epsilon)$ for any $\epsilon >0$.
\label{omegabound}
\end{lemma}
\begin{proof} Recall that for any $\epsilon>0$ and for all but finitely many $n$ we have $\omega(n) < \frac{(1+\epsilon)\log n}{\log \log n}$. As before, we will take $\epsilon=1$, and so for all but finitely many $n$, $\omega(n) < \frac{2\log n}{\log \log n}$. Let $N$ be the set of positive integers violating the prior inequality, and let $M$ be the set of ideals with norm in $N$ or with norm less than or equal to $e^e$. Note that since $N$ is finite, so is $M$. Let $C_M$ be the maximum number of distinct prime divisors of any element of $M$.

Now, for any given ideal $I$, set $I_0=rad(I)$ to the largest squarefree ideal divisor of $I$.

So we need to just estimate $\omega(I_0)$  If  $I_0 \in M$, then $\omega(I_0) \leq C_M$. If $I_0 \notin M$, then since any prime in $\textbf{Z}$ factors into at most $d$ distinct prime ideals in $K$, we have
$$\omega(I_0) \leq  \frac{2d\log \textbf{N}I_0}{\log \log \textbf{N}I_0} \leq \frac{2d \log  \textbf{N}I}{\log \log \textbf{N}I}.$$ The last inequality follows from the fact that for $x > e^e$, $\frac{\log x}{\log \log x}$ is an increasing function.

To prove the result for $\tau(I)$, note that $\tau(n)=O(n^\epsilon)$ and that it suffices to prove that there is a constant $m$, such that  $$\tau(I)= o(\textbf{N}(I)),$$ which we need to only prove for ideals whose norm is a power a prime since $\tau(I)$ is a multiplicative function.  Assume that $\textbf{N}(I)=p^a$, so that $\tau(\textbf{N}(I))=a+1$. Then the number of divisors of $I$ is bounded by the number of solutions in nonnegative integers to the inequality $$x_1 + x_2 + \cdots + x_d  \leq a+1$$ which is  $(a+d+1)!/(d!(a+1)!)$. Since $d=[K:Q]$ is fixed the result follows.
\end{proof}

\begin{theorem} For any $\alpha <3$ we have $P_K(x) = O(\frac{x^2}{(\log x)^\alpha})$.
\end{theorem}
\begin{proof} The proof is similar to our earlier estimate for $G(x)$.

We will show then the equivalent result that for any fixed $\alpha$ such that $1< \alpha < 3$, we have $P_K(x) =O(\frac{x^2 (\log \log x)^2}{(\log x)^\alpha})$.

 We will write $\mathrm{ord}(I)$ to be the smallest positive integer $o$ such that $a^o-1$ is in $I$. We note that $$P_K(x) \leq \sum_{\textbf{N}I \leq x} \frac{\textbf{N}I}{\mathrm{ord}(I)}$$ Let $c$ be the constant from Lemma \ref{idealcountlemma}. And let $\ell(x)=3\log_C x$, and set $t=(\log x)^\alpha$.

 We set $T(x,p,q)$ to be those $I$ with norm at most $x$ and satisfying $p \leq \mathrm{ord}(I) < q$

 $$P_K(x) \leq \mathrm{I}(x) + \mathrm{II}(x) + \mathrm{III}(x) $$ where we have sums defined in an analogous fashion as earlier. That is,
 \begin{equation}\label{Idef2} \mathrm{I}(x)= \sum_{I \in T(x,1,\ell(x))} \frac{{\bf{N}}I}{{\mathrm{ord}}(I)},\end{equation} \begin{equation}\label{IIdef2} \mathrm{II}(x)= \sum_{I \in T(x,\ell(x),t)} \frac{{\bf{N}}I}{{\mathrm{ord}}(I)}\end{equation} and \begin{equation}\label{IIIdef2} \mathrm{III}(x)= \sum_{I \in T(x,t, \infty) } \frac{{\bf{N}}I}{{\mathrm{ord}}(I)}.\end{equation}

We will first estimate $\mathrm{I}(x)$. The number of terms in $\mathrm{I}(x)$ is bounded by $\sum_{1 \leq r \leq \ell(x)} \tau((a^r-1))$. This sum is at most
$\ell(x)\tau_m(x)$ where $\tau_m(x)$ is the maximum of $\tau((a^r-1))$ over $r$ satisfying $1 \leq r \leq \ell(x)$. By Lemma \ref{omegabound} we have $\tau_m(I) = O(x^{1/2})$. Any term in $\mathrm{I}(x)$ is bounded above by $x$ and so conclude that $$I(x)= O(x^{1/2}3(\log_c x) x)= O(x^{3/2}\log x).$$

Next, we need to estimate $\mathrm{III}(x)$. $\mathrm{III}(x)$ has at most $O(x)$ terms (since the number of ideals of norm at most $x$ is $O(x)$) and each term is at most $\frac{x}{t}$ and so $$\mathrm{III}(x) =O\left(\frac{x^2}{t}\right)=O\left(\frac{x^2}{(\log x)^\alpha}\right).$$

To estimate $\mathrm{II}(x)$ we will break it into two sets of terms depending on how many distinct prime factors $I$ has, along with an $O(1)$ as before.

Let $k$ be a constant that satisfies Lemma \ref{omegabound}. We define $N_k$ to be the set of ideals which divide some $J$ with $\omega(J) \geq \frac{C\log \textbf{N}j}{\log \log \textbf{N}}$. Note that  $N_K$ is finite.

As before, fix $\beta=  \frac{\alpha -1}{2}$. Set $H(x)$ to be the subset
of elements of $T(x,\ell(x),t)$ with are not in $N_K$ and which have at most $(\log x)^\beta$ distinct prime factors and set  \begin{equation}\label{II1def2} \mathrm{II}_1(x) = \sum_{I \in H(x) } \frac{{\bf{N}}I}{\mathrm{ord}(I)}. \end{equation} Similarly,
let $J(x)$ to be the set of elements of $T(x,\ell(x),t)$ with more than $(\log x)^\beta$ distinct prime factors, and not in $N_K$ and set \begin{equation}\label{II2def2} \mathrm{II}_2(x) = \sum_{I \in J(x)} \frac{{\bf{N}}I}{\mathrm{ord}(I)}.\end{equation}

To estimate $\mathrm{II}_1(x)$ we will estimate $|H(x)|$. If $I \in H(x)$ then $I|((u^r-1))$ for some $r$ with $\ell(x) \leq r \leq t$
and $((a^r-1))$ not in $N_K$. Then by Lemma \ref{fieldnormbound} and Lemma \ref{omegabound}we have for some constant $M$, $$\omega((a^r-1)) < \frac{Mr}{\log r} < t.$$ The last inequality is valid for $t$ sufficiently large, which occurs when $x$ is sufficiently large.
However, each element of $|H(x)|$ has $j$ distinct prime divisors where $1 \leq j \leq (\log x)^\beta$. So for any fixed $r$, there are at most $\sum_{j=1}^m {k \choose j}$ possible choices for the distinct prime divisors where $k$ is the largest integer less than or equal to $t$
and $m$ is the largest integer less than or equal to $(\log x)^\beta$. Any prime can
be raised to at most the $\log_2 x$ power, so the total number of divisors is bounded by  $(\sum_{j=1}^m {k \choose j})(\log_2 x)^{(\log x)^\beta}$. There are at most $t$ possible values of $r$. Thus,
$$|H(x)| \leq \sum_{j=1}^m {k \choose j}t(\log_2 x)^{(\log x)^\beta}.$$   Applying Lemma \ref{HRlemma} and using the values of $k$ and $m$ then gives $$ |H(x)|\leq t\left(\frac{e^2 t }{(\log x)^\beta }\right)^{(\log x)^\beta} (\log_2 x)^{(\log x)^\beta}.$$

Now, since $t= (\log x)^\beta$, $$\log |H(x)| \leq C(\log x)^\beta(\log \log x) + \log t + (\log \log_2 x)(\log x)^\beta  $$ for some constant $C$ and this is $O((\log x)^\beta \log \log x)$. Thus $|H(x)| = O(x^{\epsilon_0})$, and so $\mathrm{II}_1(x) = O(x^{1+\epsilon_0})$ for any $\epsilon_0>0$.

To estimate  $\mathrm{II}_2(x)$, apply Lemma \ref{omegacountbound2} so that the number of terms of $J$ is $O(\frac{x (\log \log x)^2}{(\log x)^{2\beta}})$ and thus conclude that $$\mathrm{I}_2(x) = O\left(\frac{x^2 (\log \log x)^2}{(\log x)^{2\beta +1}}\right)$$ since every
term is at most $O(\frac{x}{\log_{c} x})$.

\end{proof}