Mathematical function
In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as
formula_1
for real values of "t". Here the argument is chosen in such a way that a continuous function is obtained and formula_2 holds, i.e., in the same way that the principal branch of the log-gamma function is defined.
It has an asymptotic expansion
formula_3
which is not convergent, but whose first few terms give a good approximation for formula_4. Its Taylor-series at 0 which converges for formula_5 is
 formula_6
where formula_7 denotes the polygamma function of order formula_8.
The Riemann–Siegel theta function is of interest in studying the Riemann zeta function, since it can rotate the Riemann zeta function such that it becomes the totally real valued Z function on the critical line formula_9 .
Curve discussion.
The Riemann–Siegel theta function is an odd real analytic function for real values of formula_10 with three roots at formula_11 and formula_12. It is an increasing function for formula_13, and has local extrema at formula_14, with value formula_15. It has a single inflection point at formula_16 with formula_17, which is the minimum of its derivative.
Theta as a function of a complex variable.
We have an infinite series expression for the log-gamma function
formula_18
where "γ" is Euler's constant. Substituting formula_19 for "z" and taking the imaginary part termwise gives the following series for "θ"("t")
formula_20
For values with imaginary part between −1 and 1, the arctangent function is holomorphic, and it is easily seen that the series converges uniformly on compact sets in the region with imaginary part between −1/2 and 1/2, leading to a holomorphic function on this domain. It follows that the Z function is also holomorphic in this region, which is the critical strip.
We may use the identities
formula_21
to obtain the closed-form expression
formula_22
which extends our original definition to a holomorphic function of "t". Since the principal branch of log Γ has a single branch cut along the negative real axis, "θ"("t") in this definition inherits branch cuts along the imaginary axis above "i"/2 and below −"i"/2.
Gram points.
The Riemann zeta function on the critical line can be written
formula_23
formula_24
If formula_10 is a real number, then the Z function formula_26 returns "real" values.
Hence the zeta function on the critical line will be "real" either at a zero, corresponding to formula_27, or when 
formula_28. Positive real values of formula_10 where the latter case occurs are called Gram points, after J. P. Gram, and can of course also be described as the points where formula_30 is an integer.
A Gram point is a solution formula_31 of
formula_32
These solutions are approximated by the sequence:
formula_33
where formula_34 is the Lambert W function.
Here are the smallest non negative Gram points
The choice of the index "n" is a bit crude. It is historically chosen in such a way that the index is 0 at the first value which is larger than the smallest positive zero (at imaginary part 14.13472515 ...) of the Riemann zeta function on the critical line. Notice, this formula_35-function oscillates for absolute-small real arguments and therefore is not uniquely invertible in the interval [−24,24]. Thus the odd theta-function has its symmetric Gram point with value 0 at index −3.
Gram points are useful when computing the zeros of formula_36. At a Gram point formula_37
formula_38
and if this is "positive" at "two" successive Gram points, formula_36 must have a zero in the interval.
According to Gram’s law, the real part is "usually" positive while the imaginary part alternates with the Gram points, between "positive" and "negative" values at somewhat regular intervals.
formula_40
The number of roots, formula_41, in the strip from 0 to "T", can be found by
formula_42
where formula_43 is an error term which grows asymptotically like formula_44.
Only if formula_31 would obey Gram’s law, then finding the number of roots in the strip simply becomes
formula_46
Today we know, that in the long run, Gram's law fails for about 1/4 of all Gram-intervals to contain exactly 1 zero of the Riemann zeta-function. Gram was afraid that it may fail for larger indices (the first miss is at index 126 before the 127th zero) and thus claimed this only for not too high indices. Later Hutchinson coined the phrase "Gram's law" for the (false) statement that all zeroes on the critical line would be separated by Gram points.
References.
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