Riemann–Siegel theta function
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In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as
for real values of t. Here the argument is chosen in such a way that a continuous function is obtained and holds, i.e., in the same way that the principal branch of the log-gamma function is defined.
It has an asymptotic expansion
which is not convergent, but whose first few terms give a good approximation for . Its Taylor-series at 0 which converges for is
where denotes the polygamma function of order . 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 .