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On Dirichlet Series for Sums of Squares

  • Jonathan Michael Borwein
  • Kwok-Kwong Stephen Choi
Part of the Developments in Mathematics book series (DEVM, volume 10)

Abstract

Hardy and Wright (An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979) recorded elegant closed forms for the generating functions of the divisor functions σ k (n) and σ k 2 (n) in the terms of Riemann Zeta function ζ(s) only. In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem 2.1 below, we are able to generalize the above result and prove that if f i and g i are completely multiplicative, then we have
$$ \sum\limits_{n = 1}^\infty {\frac{{({f_{1*{g_1}}})\cdot({f_{2*{g_2}}})(n)}}{{{n^s}}}} = \frac{{{L_{{f_1}{f_2}}}(s){L_{{g_1}{g_2}}}(s){L_{{f_1}{g_2}}}(s){L_{{g_1}{f_2}}}(s)}}{{{L_{{f_1}{f_2}{g_1}{g_2}}}(2s)}} $$
where \( {L_f}(s): = \sum\nolimits_{n = 1}^\infty {f(n){n^{ - s}}} \) is the Dirichlet series corresponding to f. Let r N (n) be the number of solutions of x 1 2 + … + x N 2 = n and r 2, p (n) be the number of solutions of x 2 + Py 2 = n. One of the applications of Theorem 2.1 is to obtain closed forms, in terms of ζ (s) and Dirichlet L-functions, for the generating functions of r N (n), r N 2 (n), r 2, p (n) and r 2, p (n)2 for certain N and P. We also use these generating functions to obtain asymptotic estimates of the average values for each function for which we obtain a Dirichlet series.

Key words

Dirichlet series sums of squares closed forms binary quadratic forms disjoint discriminants L-functions 

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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Jonathan Michael Borwein
    • 1
  • Kwok-Kwong Stephen Choi
    • 1
  1. 1.CECM, Department of MathematicsSimon Fraser UniversityBurnabyCanada

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