The Development of a Hybrid Asymptotic Expansion for the Hardy Function , Consisting of Just Main Sum Terms, Some Less than the Celebrated Riemann-siegel Formula

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first discovered amongst by Bessel-Hagen in 1926 and expanded upon by C. L. Siegel on his 1932 account of Riemann’s unpublished work on the zeta function.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 . 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 . 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 . Closer examination of this correlation reveals that is possible to formulate a hybrid asymptotic formula for , 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 . 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.