Abstract
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: \(\zeta (s)=\chi (s)\zeta (1-s)\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atkinson, F.V.: The mean value of the Riemann zeta-function. Acta Math. 81, 353–376 (1949)
Bettin, S., Chandee, V., Radziwiłł. M.: The mean square of the product of the Riemann zeta-function with Dirichlet polynomials. J. Reine Angew. Math. 729, 51–79 (2017)
Chandrasekharan, K.: Arithmetical Functions. Springer, New York (1970)
Hall, R.R.: The behaviour of the Riemann zeta-function on the critical line. Mathematika 46, 281–313 (1999)
Hardy, G.H., Littlewood, J.E.: Contributions to the theory of the Riemann zeta-function and the distribution of primes. Acta Math. 41, 119–196 (1918)
Hardy, G.H., Littlewood, J.E.: The approximate functional equation in the theory of the zeta-function, with applications to the divisor problems of Dirichlet and Piltz. Proc. Lond. Math. Soc. 21(2), 39–74 (1922)
Hardy, G.H., Littlewood, J.E.: The approximate functional equation of \(\zeta (s)\) and \(\zeta ^2(s)\). Proc. Lond. Math. 29(2), 81–97 (1929)
Heath-Brown, D.R.: The fourth power moment of the Riemann zeta-function. Proc. Lond. Math. Soc. 38(3), 385–422 (1979)
Ingham, A.E.: Mean value theorems in the theory of the Riemann zeta-function. Proc. Lond. Math. Soc. 27(2), 273–300 (1926)
Ivić, A.: The Theory of the Riemann Zeta-Function. Wiley, New York (1985) (2nd ed. Dover, Mineora, (2003))
Ivić, A.: Lectures on Mean Values of the Riemann Zeta-Funtion, Tata Inst. Fund. Res. Lec. Math. Phy. 82, Bombay (1991)
Ivić, A.: On the integral of Hardy’s function. Arch. Math. 83, 41–47 (2004)
Ivić, A.: On the divisor function and the Rimann zeta-function in short intervals. Ramanujan J. 19, 207–224 (2009)
Ivić, A.: The Theory of Hardy’s \(Z\)-Function. Cambridge University Press, Cambridge (2013)
Ivić, A.: On a cubic moment of Hardy’s function with a shift. In: Montgomery, H.L., Nikeghbali, A., Rassias, M.T. (eds.) Exploring the Riemann Zeta Function, 99–112. Springer, Berlin (2017)
Ivić, A., Zhai, W.: On certain integrals involving the Dirichlet divisor problem. Functiones et Approximatio 62(2), 247–267 (2020)
Jutila, M.: Atkinson’s formula for Hardy’s function. J. Number Theory 129, 2853–2878 (2009)
Jutila, M.: An asymptotic formula for the primitive of Hardy’s function. Ark. Math. 49, 97–107 (2011)
Karatsuba, A.A., Vononin, S.M.: The Riemann Zeta-Function. Walter de Gruiter, Berlin (1992)
Korolev, M.A.: On the integral of Hardy’s function. Izv. Math. 72, 429–478 (2008)
Motohashi, Y.: Spectral Theory of the Riemann Zeta-Function. Cambridge University Press, Cambridge (1997)
Robert, O., Sargos, P.: Three-dimensional exponential sums with monomials. J. Reine Angew. Math. 591, 1–20 (2006)
Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford Uiversity Press, Oxford (1986)
Authors' contributions
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by the National Natural Science Foundation of China(Grant No. 11971476)
Rights and permissions
About this article
Cite this article
Cao, X., Tanigawa, Y. & Zhai, W. Some mean value results related to Hardy’s function. Res. number theory 7, 30 (2021). https://doi.org/10.1007/s40993-021-00255-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-021-00255-z