Editing Riemann–Siegel theta function (section)

{{Short description|Mathematical function}}
In [[mathematics]], the '''Riemann–Siegel theta function''' is defined in terms of the [[gamma function]] as

:<math>\theta(t) = \arg \left( \Gamma\left(\frac{1}{4}+\frac{it}{2}\right) \right) - \frac{\log \pi}{2} t</math>

for real values of&nbsp;''t''.  Here the [[argument (complex analysis)|argument]] is chosen in such a way that a continuous function is obtained and <math>\theta(0)=0</math> holds, i.e., in the same way that the [[principal branch]] of the [[Gamma function#The log-gamma function|log-gamma]] function is defined.

It has an [[asymptotic expansion]]

:<math>\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</math>

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

: <math>\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}</math>

where <math>\psi^{(2k)}</math> denotes the [[polygamma function]] of order <math>2k</math>.
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 (mathematics)|critical line]] <math>s = 1/2 + i t</math> .