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

Author

Listed:
  • 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)

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 .

Suggested Citation

  • Jing Huang & Yukun Liu & Deyu Zhang, 2025. "On Some Mean Value Results for the Zeta-Function and a Rankin–Selberg Problem," Mathematics, MDPI, vol. 13(16), pages 1-16, August.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:16:p:2681-:d:1728761
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