A computational algorthm for the Hardy function Z(t), utlising sub-sequences of generalised quadratic Gauss sums, with an overall operational complexity O(((t/epsilon)^(1/3))*(log(t))^(2+o(1))))

TL;DR

This paper introduces an efficient computational method for the Hardy function Z(t) using quadratic Gaussian sums, achieving significantly lower complexity than classical methods, with practical implementation and promising future improvements.

Contribution

The paper develops a novel algorithm for computing quadratic Gaussian sums efficiently and integrates it into a new method for calculating Z(t) with reduced operational complexity.

Findings

Algorithm ZT13 outperforms classical methods in efficiency.

Sample computations confirm theoretical complexity and accuracy.

Potential for further computational savings is discussed.

Abstract

This paper describes a practical methodology for computing the Hardy function Z(t), using just O(((t/epsilon)^(1/3))*(log(t))^(2+o(1)))) standard computational operations, to a tolerance of epsilon in the relative error. The methodology is analogous to work published relatively recently by G. H. Hiary, although the details are very different. Initially the Hardy function is formulated into sub-sequences of generalised quadratic Gaussian sums of ever increasing length N. Adapting a theoretical framework formulated by R. B. Paris, an algorithm for the computation of quadratic Gaussian sums using just O(log(N)) operations is developed and tested. This algorithmic methodology is itself incorporated (as a sub-program) into a somewhat larger algorithm (algorithm ZT13) designed for the computation of Z(t), which is considerably more efficient than classical O(sqrt(t)) methods, as exemplified by variants of the Riemann-Siegel formula. Sample computations using algorithm ZT13 are presented, the results of which conform to technical theoretical predictions made regarding the limits of its performance (both in terms of its operational count and degree of precision). Speculative ideas briefly floated in the conclusions suggest that adaptations to this methodology have the potential to deliver significant further computational savings in the future.