Some mean value results related to Hardy’s function


Let \(\zeta (s)\) and Z(t) be the Riemann zeta function and Hardy’s function respectively. We show asymptotic formulas for \(\int _0^T Z(t)\zeta (1/2+it)dt\) and \(\int _0^T Z^2(t) \zeta (1/2+it)dt\). Furthermore we derive an upper bound for \(\int _0^T Z^3(t) \chi ^{\alpha }(1/2+it)dt\) for \(-1/2<\alpha <1/2\), where \(\chi (s)\) is the function which appears in the functional equation of the Riemann zeta function: \(\zeta (s)=\chi (s)\zeta (1-s)\).

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Authors' contributions

The authors are very grateful to the referee for his/her valuable comments and suggestions. Furthermore he/she kindly pointed out the authors the reference [2], where (7) is established, and the possibility to prove Theorem 3 more quickly from starting with \(\int _0^T|\zeta (1/2+it)|^2n^{-it}dt\).

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Correspondence to Yoshio Tanigawa.

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This work is supported by the National Natural Science Foundation of China(Grant No. 11971476)

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Cao, X., Tanigawa, Y. & Zhai, W. Some mean value results related to Hardy’s function. Res. number theory 7, 30 (2021).

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  • Hardy’s function
  • Mean value theorems
  • Approximate functional equation
  • Exponential sum and integral

Mathematics Subject Classification

  • 11M06
  • 11N07