Riemann-Siegel theta function

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 ${\displaystyle \theta (0)=0}$ holds, i.e., in the same way that the principal branch of the log Gamma function is defined.

It has an asymptotic expansion

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

...
Wikipedia