Riemann-Siegel theta function

In mathematics, the Riemann-Siegel theta function is the function

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

It has an asymptotic formula which gives very accurate results for real values of t which are at all large

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

The interest in the Riemann-Siegel theta function is in studying the Riemann zeta function and defining the Z function.

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

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

where γ is Euler's constant. Substituting $\frac{2it+1}{4}$ for z and taking the imaginary part termwise gives the following series for θ(t)

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

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.

Gram points

The Riemann zeta function on the critical line can be written

$\zeta(\frac{1}{2}+it) = \exp(-i \theta(t))Z(t)$ ,

where Z, the Z function, is real for real t. Hence the zeta function on the critical line will be real when $sin(θ(t)) = 0$ . 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.

Gram points are useful when computing the zeros of Z(t). At a Gram point g_n, $\zeta(\frac{1}{2}+ig_n) = \cos(\theta(g_n))Z(g_n) = (-1)^n Z(g_n)$ , and if this is positive at two successive Gram points, Z must have a zero in the interval. Since (-1)ⁿZ(g_n) very often is positive (a phenomenon called "Gram's law") this turns out in practice to be quite useful.