Computing the truncated theta function via Mordell integral

Abstract:

Hiary has presented an algorithm which allows us to evaluate the truncated theta function $\sum _{k=0}^n \exp (2\pi \mathrm {i} (zk+\tau k^2))$ to within $\pm \epsilon$ in $O(\ln (\tfrac {n}{\epsilon })^{\kappa })$ arithmetic operations for any real $z$ and $\tau$. This remarkable result has many applications in Number Theory, in particular, it is the crucial element in Hiary’s algorithm for computing $\zeta (\tfrac {1}{2}+\mathrm {i} t)$ to within $\pm t^{-\lambda }$ in $O_{\lambda }(t^{\frac {1}{3}}\ln (t)^{\kappa })$ arithmetic operations. We present a significant simplification of Hiary’s algorithm for evaluating the truncated theta function. Our method avoids the use of the Poisson summation formula, and substitutes it with an explicit identity involving the Mordell integral. This results in an algorithm which is efficient, conceptually simple and easy to implement.