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1997 | 82 | 4 | 309-330
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New integral representations for the square of the Riemann zeta-function

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Introduction. The recent discovery of an analogue of the Riemann-Siegel integral formula for Dirichlet series associated with cusp forms [2] naturally raises the question whether similar formulas might exist for other types of zeta functions. The proof of these formulas depends on the functional equation for the underlying Dirichlet series. In both cases, for ζ(s) and for the cusp form zeta functions, only a simple gamma factor is involved. The next simplest case arises when two such factors occur in the functional equation. The prototype of these Dirichlet series is ζ²(s), and so any investigation might well begin with this example. In the present study we show that, indeed, a formula of Riemann-Siegel type can be found for ζ²(s). The numerous applications of the ordinary Riemann-Siegel integral formula [3] suggest similar ones for our formula too. For instance, it seems very probable to derive an asymptotic expansion for ζ²(s), giving a generalization of Siegel's result [9]. Originally, this expansion is due to Motohashi [6,7], and it depends on the corresponding formula for ζ(s). Consequently, our approach might lead to an independent proof of Motohashi's expansion. This would be of considerable value, since our method applies as well to other Dirichlet series satisfying similar functional equations, like Hecke L series of quadratic fields. As a first step in this direction we give a simple proof of the approximate functional equation for ζ²(s) at the end of the paper. The author is very much indebted to Professor Aleksandar Ivić (Belgrade) who read a preliminary version of this work. His suggestions and criticism led to numerous improvements. Thanks are also due to the referee for pointing out some misprints and unclear passages.
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  • Fachbereich Mathematik, Universität Kaiserslautern, D-67663 Kaiserslautern, Germany
  • [1] T. Estermann, On the representation of a number as the sum of two products, Proc. London Math. Soc. (2) 31 (1930), 123-133.
  • [2] A. Guthmann, The Riemann-Siegel integral formula for Dirichlet series associated to cusp forms, in: Analytic and Elementary Number Theory, Vienna, 1996, 53-69.
  • [3] A. Ivić, The Riemann Zeta Function, Wiley, New York, 1985.
  • [4] T. Kubota, Elementary Theory of Eisenstein Series, Kodansha, Tokyo, 1973.
  • [5] N. N. Lebedev, Special Functions and their Applications, Dover, New York, 1972.
  • [6] Y. A. Motohashi, Note on the approximate functional equation for ζ²(s) I, Proc. Japan Acad. Ser. A 59 (1983), 393-396; II, Proc. Japan Acad. Ser. A. 59 (1983), 469-472; III, Proc. Japan Acad. Ser. A. 62 (1986), 410-412.
  • [7] Y. A. Motohashi, An asymptotic expansion of the square of the Riemann zeta-function, in: Sieve Methods, Exponential Sums, and their Applications in Number Theory (Cardiff, 1995), Cambridge Univ. Press, Cambridge, 1997, 293-307.
  • [8] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.
  • [9] C. L. Siegel, Über Riemanns Nachlaß zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik 2 (1932), 45-80.
  • [10] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1944.
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