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arXiv · 1511.07140

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

Aleksandar Ivić
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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 $$ Z(t) := ζ(1/2+it){\bigl(χ(1/2+it)\bigr)}^{-1/2}\quad(t\in\Bbb R), \quad ζ(s) = χ(s)ζ(1-s) $$ 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 $$ \int_1^TZ^3(t)\,dt \;=\;O_\varepsilon(T^{3/4+\varepsilon}). $$

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Aleksandar Ivić. 2015-11-23. On a cubic moment of Hardy's function with a shift. https://arxiv.org/abs/1511.07140

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