Riemann-Siegel theta function

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In mathematics, the Riemann-Siegel theta function is defined in terms of the Gamma function as

$\theta(t) = \arg \left( \Gamma\left(\frac{2it+1}{4}\right) \right) - \frac{\log \pi}{2} t$

for real values of t. Here the argument is chosen in such a way that a continuous function is obtained, i.e., in the same way that the principal branch of the log Gamma function is defined.

It has an asymptotic expansion

$\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots$

which is strongly convergent for $t \gg 1$ .

It is of interest in studying the Riemann zeta function, since it gives the argument of the zeta function on the critical line $s = 1 / 2 + i t$ .

The Riemann-Siegel theta function is an odd real analytic function for real values of t; it is an increasing function for values |t| > 6.29.

Theta as a function of a complex variable

We have an infinite series expression for the log Gamma function

$\log \Gamma z = -\gamma z -\log z + \sum_{n=1}^\infty \left(\frac{z}{n} - \log \left(1+\frac{z}{n}\right)\right),$

where ? is Euler's constant. Substituting $(2 i t + 1) / 4$ for z and taking the imaginary part termwise gives the following series for ?(t)

$\theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t + \sum_{n=1}^\infty \left(\frac{t}{2n} - \arctan\left(\frac{2t}{4n+1}\right)\right)$

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

$\arg z = \frac{\log z - \log\bar z}{2i}\quad\text{and}\quad\overline{\Gamma(z)}=\Gamma(\bar z)$

to obtain the closed-form expression

$\theta(t) = \frac{\log\Gamma\left(\frac{2it+1}{4}\right)-\log\Gamma\left(\frac{-2it+1}{4}\right)}{2i} - \frac{\log \pi}{2} t,$

which extends our original definition to a holomorphic function of t. Since the principal branch of log G has a single branch cut along the negative real axis, ?(t) in this definition inherits branch cuts along the imaginary axis above i/2 and below -i/2.

**Riemann-Siegel theta function in the complex plane**

$-1 < \Re(t) < 1$	$-5 < \Re(t) < 5$	$-40 < \Re(t) < 40$

Gram points

The Riemann zeta function on the critical line can be written

$\zeta\left(\frac{1}{2}+it\right) = e^{-i \theta(t)}Z(t),$

$Z(t) = e^{i \theta(t)} \zeta\left(\frac{1}{2}+it\right)$

If t is a real number, then the Z function, $Z\left(t\right)$ , returns real values.

Hence the zeta function on the critical line will be real when $\sin\left(\,\theta(t)\,\right)=0$ . Positive real values of t where this occurs are called Gram points, after J.-P. Gram, and can of course also be described as the points where $\frac{\theta(t)}{\pi}$ is an integer.

A Gram point is a solution, n of

$\theta\left(g_{n}\right) = n\pi$

Here are some examples of Gram points

$n$	$g n$
0	17.8455995404
1	23.1702827012
2	27.6701822178

Gram points are useful when computing the zeros of $Z\left(t\right)$ . At a Gram point $g n$ ,

$\zeta\left(\frac{1}{2}+ig_n\right) = \cos(\theta(g_n))Z(g_n) = (-1)^n Z(g_n),$

and if this is positive at two successive Gram points, $Z\left(t\right)$ must have a zero in the interval.

According to Gram’s law, the real part is usually positive^{[dubious – discuss]} while the imaginary part alternates with the gram points, between positive and negative values at somewhat regular intervals.

$\Re\left\{\,(-1)^n \, Z\left(g_{n}\right)\,\right\} > 0$

The number of roots, $R\left(t\right)$ , in the strip from 0 to t, can be found by

$R\left(t\right) = \frac{\theta(t)}{\pi} + 1$

If $g n$ obeys Gram’s law, then finding the number of roots in the strip simply becomes

$R\left(g_{n}\right) = n + 1$

Category: Zeta and L-functions

Hidden categories: All pages needing cleanup | Articles with disputed statements from November 2008

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