Proceedings of the Steklov Institute of Mathematics

, Volume 299, Issue 1, pp 109–116 | Cite as

An Approximate Functional Equation for the Primitive of Hardy’s Function

  • Matti Jutila


A formula of Atkinson type for the primitive of Hardy’s function is generalized to the case where the lengths of the two sums involved in that formula vary in wide ranges.


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  1. 1.
    F. V. Atkinson, “The mean-value of the Riemann zeta function,” Acta Math. 81, 353–376 (1949).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    J. Bourgain, “Decoupling, exponential sums and the Riemann zeta function,” J. Am. Math. Soc. 30 (1), 205–224 (2017).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    M. N. Huxley, Area, Lattice Points, and Exponential Sums (Clarendon Press, Oxford, 1996).MATHGoogle Scholar
  4. 4.
    A. Ivić, The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications (J. Wiley & Sons, New York, 1985); reprinted as The Riemann Zeta-Function: Theory and Applications (Dover Publ., Mineola, NY, 2003).MATHGoogle Scholar
  5. 5.
    A. Ivić, The Theory of Hardy’s Z-Function (Cambridge Univ. Press, Cambridge, 2013).MATHGoogle Scholar
  6. 6.
    M. Jutila, “Transformation formulae for Dirichlet polynomials,” J. Number Theory 18, 135–156 (1984).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    M. Jutila, Lectures on a Method in the Theory of Exponential Sums (Springer, Berlin, 1987), Tata Inst. Fundam. Res., Tata Lect. Math. Phys. 80.MATHGoogle Scholar
  8. 8.
    M. Jutila, “Atkinson’s formula for Hardy’s function,” J. Number Theory 129 (11), 2853–2878 (2009).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    M. Jutila, “Transformations of zeta-sums,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 276, 155–161 (2012) [Proc. Steklov Inst. Math. 276, 149–155 (2012)].MathSciNetMATHGoogle Scholar
  10. 10.
    A. A. Karatsuba, “On the distance between adjacent zeros of the Riemann zeta function lying on the critical line,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 157, 49–63 (1981) [Proc. Steklov Inst. Math. 157, 51–66 (1983)].MathSciNetMATHGoogle Scholar
  11. 11.
    M. A. Korolev, “On the integral of Hardy’s function Z(t),” Izv. Ross. Akad. Nauk, Ser. Mat. 72 (3), 19–68 (2008) [Izv. Math. 72, 429–478 (2008)].MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., rev. by D. R. Heath-Brown (Clarendon Press, Oxford, 1986).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

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