The Ramanujan Journal

, Volume 7, Issue 1–3, pp 95–127

# On Dirichlet Series for Sums of Squares

• Jonathan Michael Borwein
• Kwok-Kwong Stephen Choi
Article

## 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 fi and gi are completely multiplicative, then we have$$\sum\limits_{n = 1}^\infty {\frac{{(f_1 * g_1 )(n) \cdot (f_2 * g_2 )(n)}}{{n^s }} = } \frac{{L_{f_1 f_2 } (s)L_{g_1 g_2 } (s)L_{f_1 g_2 } (s)L_{g_1 f_2 } (s)}}{{L_{f_1 f_2 g_1 g_2 } (2s)}}$$where Lf(s) := ∑n = 1f(n)n−s is the Dirichlet series corresponding to f. Let rN(n) be the number of solutions of x12 + ··· + xN2 = n and r2,P(n) be the number of solutions of x2 + Py2 = 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 rN(n), rN2(n), r2,P(n) and r2,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.

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

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