On Dirichlet Series for Sums of Squares

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 _k(n) and _k²(n) in the terms of Riemann Zeta function (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 havewhere L_f(s) := _{n = 1}f(n)n^–s is the Dirichlet series corresponding to f. Let r_N(n) be the number of solutions of x₁² + ··· + x_N² = n and r_2,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 (s) and Dirichlet L-functions, for the generating functions of r_N(n), r_N²(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.