On a cubic moment of Hardy's function with a shift

TL;DR

This paper derives an asymptotic formula for a shifted cubic moment of Hardy's function and discusses its implications, supporting a conjecture about the growth rate of the cubic moment of Z(t).

Contribution

It provides a new asymptotic formula for a shifted cubic moment of Hardy's function and offers evidence supporting a conjecture on its growth rate.

Findings

Derived an asymptotic formula for the shifted cubic moment of Hardy's function.

Presented a mean value result supporting the conjecture that the cubic moment grows at most on the order of T^{3/4+ε}.

Discussed the cubic moment of Hardy's function and its implications.

Abstract

An asymptotic formula for $$ \int_{T/2}^{T}Z^2(t)Z(t+U)\,dt\qquad(0< U = U(T) \le T^{1/2-\varepsilon}) $$ is derived, where $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}\quad(t\in\Bbb R), \quad \zeta(s) = \chi(s)\zeta(1-s) $$ is Hardy's function. The cubic moment of $Z(t)$ is also discussed, and a mean value result is presented which supports the author's conjecture that $$ \int_1^TZ^3(t)\,dt \;=\;O_\varepsilon(T^{3/4+\varepsilon}). $$