On Some Mean Value Results for the Zeta-Function and a Rankin–Selberg Problem

Jing Huang, Yukun Liu and Deyu Zhang ()
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Jing Huang: School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
Yukun Liu: School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
Deyu Zhang: School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China

Mathematics, 2025, vol. 13, issue 16, 1-16

Abstract: Denote by Δ 1 ( x ; φ ) the error term in the classical Rankin–Selberg problem. Denote by ζ ( s ) the Riemann zeta-function. We establish an upper bound for this integral ∫ 0 T Δ 1 ( t ; φ ) ζ 1 2 + i t 2 d t . In addition, when 2 ≤ k ≤ 4 is a fixed integer, we will derive an asymptotic formula for the integral ∫ 1 T Δ 1 k ( t ; φ ) ζ 1 2 + i t 2 d t . The results rely on the power moments of Δ 1 ( t ; φ ) and E ( t ) , the classical error term in the asymptotic formula for the mean square of ζ 1 2 + i t .

Keywords: the Rankin–Selberg problem; Riemann zeta-function; integral of the error term; mean value estimates (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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