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On a cubic moment of Hardy's function with a shift (1511.07140v1)

Published 23 Nov 2015 in math.NT

Abstract: An asymptotic formula for $$ \int_{T/2}{T}Z2(t)Z(t+U)\,dt\qquad(0< U = U(T) \le T{1/2-\varepsilon}) $$ 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 $$ \int_1TZ3(t)\,dt \;=\;O_\varepsilon(T{3/4+\varepsilon}). $$

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