Riemann–Siegel theta function

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

\theta(t) = \arg \left( \Gamma\left(\frac{2it%2B1}{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}%2B\frac{1}{48t}%2B \frac{7}{5760t^3}%2B\cdots

which is not convergent, but whose first few terms give a good approximation 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 %2B 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 \left(z\right) = -\gamma z -\log z %2B \sum_{n=1}^\infty \left(\frac{z}{n} - \log \left(1%2B\frac{z}{n}\right)\right),

where γ is Euler's constant. Substituting (2it%2B1)/4 for z and taking the imaginary part termwise gives the following series for θ(t)

\theta(t) = -\frac{\gamma %2B \log \pi}{2}t - \arctan 2t %2B \sum_{n=1}^\infty \left(\frac{t}{2n} - \arctan\left(\frac{2t}{4n%2B1}\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%2B1}{4}\right)-\log\Gamma\left(\frac{-2it%2B1}{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

Gram points

The Riemann zeta function on the critical line can be written

\zeta\left(\frac{1}{2}%2Bit\right) = e^{-i \theta(t)}Z(t),
Z(t) = e^{i \theta(t)} \zeta\left(\frac{1}{2}%2Bit\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 g_{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}%2Big_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 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} %2B 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 %2B 1.

References