Computing the truncated theta function via Mordell integral

TL;DR

This paper introduces a simplified, efficient algorithm for computing the truncated theta function using Mordell integrals, improving conceptual clarity and ease of implementation over previous methods.

Contribution

The authors present a new algorithm that replaces Poisson summation with Mordell integral identities, simplifying the evaluation of the truncated theta function.

Findings

Algorithm avoids Poisson summation formula

Achieves efficient computation of truncated theta function

Simplifies implementation and understanding

Abstract

Hiary [3] has presented an algorithm which allows to evaluate the truncated theta function $\sum_{k=0}^n \exp(2\pi \i (zk+\tau k^2))$ 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 $\zeta(\tfrac{1}{2}+\i t)$ to within $\pm t^{-\lambda}$ in $O_{\lambda}(t^{\frac{1}{3}}\ln(t)^{\kappa})$ arithmetic operations, see [2]. 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.