Abstract
Let Z(t) be the classical Hardy function in the theory of Riemann’s zeta-function. An asymptotic formula with an error term O(T 1/6log T) is given for the integral of Z(t) over the interval [0,T], with special attention paid to the critical cases when the fractional part of \(\sqrt{T/2\pi }\) is close to \(\frac{1}{4}\) or \(\frac{3}{4}\).
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Jutila, M. An asymptotic formula for the primitive of Hardy’s function. Ark Mat 49, 97–107 (2011). https://doi.org/10.1007/s11512-010-0122-4
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DOI: https://doi.org/10.1007/s11512-010-0122-4