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On a Cubic Moment of Hardy’s Function with a Shift

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Exploring the Riemann Zeta Function

Abstract

An asymptotic formula for

$$\displaystyle{ \int _{T/2}^{T}Z^{2}(t)Z(t + U)dt\qquad (0 <U = U(T)\leqslant T^{1/2-\varepsilon }) }$$

is derived, where

$$\displaystyle{ Z(t):= \zeta \left ({1 \over 2} + it\right )\big(\chi \left ({1 \over 2} + it\right )\big)^{-1/2}\quad (t \in \mathbb{R}),\quad \zeta (s) =\chi (s)\zeta (1 - s) }$$

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

$$\displaystyle{ \int _{1}^{T}Z^{3}(t)dt\; =\; O_{\varepsilon }(T^{3/4+\varepsilon }). }$$

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References

  1. S. Bettin, The second moment of the Riemann zeta function with unbounded shifts. Int. J. Number Theory 6(8), 1933–1944 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Bettin, V. Chandee, M. Radziwiłł, The mean square of the product of the Riemann zeta function with Dirichlet polynomials. J Reine Angew Math. doi:10.1515/crelle-2014-0133

    Google Scholar 

  3. R.R. Hall, A new unconditional result about large spaces between zeta zeros. Mathematika 53, 101–113 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. A.E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function. Proc. Lond. Math. Soc. (2) 27, 273–300 (1928)

    Google Scholar 

  5. A. Ivić, The Riemann Zeta-Function (Wiley, New York, 1985). 2nd edn., Dover, Mineola, New York, 2003

    Google Scholar 

  6. A. Ivić, Mean Values of the Riemann Zeta-Function, LN’s, vol. 82 (Tata Institute of Fundamental Research, Bombay, 1991). Distr. by Springer, Berlin etc.

    Google Scholar 

  7. A. Ivić, On the integral of Hardy’s function. Arch. Math. 83, 41–47 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Ivić, The Theory of Hardy’s Z-Function (Cambridge University Press, Cambridge, 2012), 245 pp.

    MATH  Google Scholar 

  9. A. Ivić, Y. Motohashi, On the fourth power moment of the Riemann zeta-function. J. Number Theory 51, 16–45 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Jutila, Atkinson’s formula for Hardy’s function. J. Number Theory 129(11), 2853–2878 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Jutila, An asymptotic formula for the primitive of Hardy’s function. Ark. Mat. 49, 97–107 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Jutila, The mean value of Hardy’s function in short intervals. Indagationes Math. (special vol. dedicated to 125th anniversary of J.G. van der Corput) 26, 867–882 (2015)

    Google Scholar 

  13. A.A. Karatsuba, S.M. Voronin, The Riemann Zeta-Function (Walter de Gruyter, Berlin/New York, 1992)

    Book  MATH  Google Scholar 

  14. M.A. Korolev, On the integral of Hardy’s function Z(t). Izv. Math. 72(3), 429–478 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72(3), 19–68 (2008)

    Google Scholar 

  15. K. Ramachandra, On the Mean-Value and Omega-Theorems for the Riemann Zeta-Function, LN’s, vol. 85 (Tata Inst. of Fundamental Research, Bombay, 1995). Distr. by Springer, Berlin etc.

    Google Scholar 

  16. S. Shimomura, Fourth moment of the Riemann Zeta-function with a shift along the real line. Tokyo J. Math. 36, 355–377 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd edn. (Oxford University Press, Oxford, 1986)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Katedra Matematike RGF-a Universiteta u Beogradu, Đjušina 7, 11000, Beograd, Serbia

    Aleksandar Ivić

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  1. Aleksandar Ivić
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Correspondence to Aleksandar Ivić .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA

    Hugh Montgomery

  2. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Ashkan Nikeghbali

  3. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Ivić, A. (2017). On a Cubic Moment of Hardy’s Function with a Shift. In: Montgomery, H., Nikeghbali, A., Rassias, M. (eds) Exploring the Riemann Zeta Function. Springer, Cham. https://doi.org/10.1007/978-3-319-59969-4_6

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