Fast methods to compute the Riemann zeta function

TL;DR

This paper introduces three new fast algorithms for computing the Riemann zeta function on the critical line, improving computational complexity over existing methods with exponents of 2/5, 1/3, and approximately 0.307.

Contribution

The paper presents three novel algorithms for zeta function computation, utilizing quadratic and cubic exponential sums with improved complexity exponents.

Findings

New method with complexity exponent 2/5

Quadratic exponential sum algorithm with exponent 1/3

Cubic exponential sum algorithm with exponent 4/13

Abstract

The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Sch\"onhage's method, or Heath-Brown's method. The complexities of these methods have exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this paper, three new fast and potentially practical methods to compute zeta are presented. One method is very simple. Its complexity has exponent 2/5. A second method relies on this author's algorithm to compute quadratic exponential sums. Its complexity has exponent 1/3. The third method employs an algorithm, developed in this paper, to compute cubic exponential sums. Its complexity has exponent 4/13 (approximately, 0.307).