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 sigma(k) (n) and sigma(2)(k)(n) in the terms of Riemann Zeta function zeta(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 [GRAPHICS] where L-f (s):= Sigma(n=1)(infinity) 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 zeta(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.