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 + it.

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.

[edit] 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 γ is Euler's constant. Substituting (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 \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 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.

Riemann-Siegel theta function in the complex plane
-1 < \Re(t) < 1 -5 < \Re(t) < 5 -40 < \Re(t) < 40

[edit] 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),

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 θ(t) / π is an integer.

Gram points are useful when computing the zeros of Z(t). At a Gram point gn,

\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 must have a zero in the interval. Since (-1)nZ(gn) very often is positive (a phenomenon called Gram's law) this turns out in practice to be quite useful.