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whichfirst discovered amongst by Bessel-Hagen in 1926 and expanded upon by C. L.Siegel on his 1932 account of Riemanns unpublished work on the zeta function.By application of standard asymptotic methods for integral estimation, and theuse of certain approximations pertaining to special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

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