A computational algorithm for the Hardy function $Z(t)$, utilising sub-sequences of generalised cubic Gauss sums, with an overall operational complexity of $O\bigl((t/\varepsilon_t)^{[.25,.3]}(\log t)^{2+o(1)}\bigr)$, for $t \in [10^{23},10^{35}]$

Abstract

In 2011 G. A. Hiary devised a computational algorithm for the Hardy function $Z(t)$, requiring just $O(t^{1/3} (\log(t))^\kappa)$ operations. This compares to $O(\sqrt t)$ operations necessary for computing $Z(t)$ using the classical Riemann-Siegel formula. The methodology involved the sub-division of the Riemann-Siegel formula into sequences of quadratic Gauss/exponential sums of various lengths $N$. Such sums can be computed rapidly, in order $\log(N)$ operations, using standard recursive schemes. More recently, the principal author developed a similar algorithm with an $O((t/\varepsilon_t)^{1/3} (\log(t))^2)$ operational count, accurate to $\epsilon_t$ in the relative error. Although constructively analogous, the sub-division into quadratic sums was applied to a different asymptotic formula for $Z(t)$, giving the new algorithm an original formulation. This paper presents a significant extension of these ideas. The main theoretical result is an asymptotic expression for $Z(t)$ in terms of sub-sequences of generalised, $m^{\rm th}$-order, Gauss sums of progressively increasing length. Computationally, the main focus falls upon the cubic Gauss sum formulation. The particular parameterisation of these cubic sums makes them amenable to rapid computation, utilising a recursive scheme similar to those implemented for quadratic sums. The net result is a computational algorithm for $Z(t)$ with a reduced $O\bigl((t/\varepsilon_t)^{[.25,.3]}(\log t)^{2+o(1)}\bigr)$ operational count for $t \in [10^{23},10^{35}]$, to high accuracy. Sample computations lend practical support to these findings.