Riemann–Siegel theta function

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

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

for real values of t. Here the argument is chosen in such a way that a continuous function is obtained and ${\displaystyle \theta (0)=0}$ holds, i.e., in the same way that the principal branch of the log-gamma function is defined.

It has an asymptotic expansion

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

which is not convergent, but whose first few terms give a good approximation for ${\displaystyle t\gg 1}$. Its Taylor-series at 0 which converges for ${\displaystyle |t|<1/2}$ is

${\displaystyle \theta (t)=-{\frac {t}{2}}\log \pi +\sum _{k=0}^{\infty }{\frac {(-1)^{k}\psi ^{(2k)}\left({\frac {1}{4}}\right)}{(2k+1)!}}\left({\frac {t}{2}}\right)^{2k+1}}$

where ${\displaystyle \psi ^{(2k)}}$ denotes the polygamma function of order ${\displaystyle 2k}$. 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 ${\displaystyle s=1/2+it}$ .