An asymptotic formula for the primitive of Hardy’s function

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 $\frac{3}{4}$.