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

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))))