The Development of a Hybrid Asymptotic Expansion for the Hardy Fuction Z(t), Consisting of Just [2*sqrt(2)-2]*sqrt(t/(2*pi)) Main Sum Terms, some 17% less than the celebrated Riemann-Siegel Formula

TL;DR

This paper introduces a hybrid asymptotic expansion for the Hardy Z(t) function, combining Riemann-Siegel and new zeta-sum terms, reducing computational time by 15% and opening new theoretical research directions.

Contribution

It develops a novel hybrid asymptotic formula for Z(t) that integrates a new zeta-sum with the Riemann-Siegel formula, improving computational efficiency.

Findings

Reduces CPU time for Z(t) calculation by 14-15%.

Derives a new zeta-sum from asymptotic analysis of the Riemann-Siegel integral.

Highlights potential for new theoretical insights into the zeta function.

Abstract

This paper begins with a re-examination of the Riemann-Siegel Integral, which first discovered amongst by Bessel-Hagen in 1926 and expanded upon by C. L. Siegel on his 1932 account of Riemanns unpublished work on the zeta function. By application of standard asymptotic methods for integral estimation, and the use of certain approximations pertaining to special functions, it proves possible to derive a new zeta-sum for the Hardy function Z(t). In itself this new zeta-sum (whose terms made up of elementary functions, but are unlike those that arise from the analytic continuation of the Dirichlet series) proves to be a computationally inefficient method for calculation of Z(t). However, by further, independent analysis, it proves possible to correlate the terms the new zeta-sum with the terms of the Riemann-Siegel formula, thought, since its discovery by Siegel, to be the most efficient means of calculating Z(t). Closer examination of this correlation reveals that is possible to formulate a hybrid asymptotic formula for Z(t) consisting of a sum containing both Riemann-Siegel terms and terms from the new zeta-sum, in such a way as to reduce the overall CPU time required by a factor between 14-15 percent. Alongside the obvious computational benefits of such a result, the very existence of the new zeta-sum itself highlights new theoretical avenues of study in this field.