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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 θ = 0 {\displaystyle \theta =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 t ≫ 1 {\displaystyle t\gg 1}. Its Taylor-series at 0 which converges for | t | < 1 / 2 {\displaystyle |t|<1/2} is
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