On Dirichlet series for sums of squares

Borwein, Jonathan M. and Choi, Stephen (2003) On Dirichlet series for sums of squares. Ramanujan Journal, 7 . pp. 95-128.

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In \cite{HW}, Hardy and Wright recorded elegant closed forms for the generating functions of the divisor functions $\sigma_k(n)$ and $\sigma_k(n)^2$:$\sum_{n=1}^\infty \frac{\sigma_k(n)}{n^s}=\zeta (s)\zeta (s-k)$ and $\sum_{n=1}^\infty \frac{\sigma_k(n)^2}{n^s}=\frac{\zeta (s)\zeta (s-k)^2\zeta (s-2k)}{\zeta (2s-2k)}.$ In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem \ref{thm 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$\sum_{n=1}^\infty \frac{(f_1\ast g_1)(n)\cdot (f_2\ast g_2)(n)}{n^s}=\frac{L_{f_1f_2}(s)L_{g_1g_2}(s)L_{f_1g_2}(s)L_{g_1f_2}(s)}{L_{f_1f_2g_1g_2}(2s)}$ where $L_f(s):=\sum_{n=1}^\infty f(n)n^{-s}$ is the Dirichlet series corresponding to $f$. Let $r_N(n)$ be the number of solutions of $x_1^2+\cdots +x_N^2=n$ and $r_{2,P}(n)$ be the number of solutions of $x^2+Py^2=n$. One of the applications of Theorem \ref{thm 2.1} is to obtain closed forms, in terms of $\zeta (s)$ and Dirichlet $L$-functions, for the generating functions of $r_N(n), r_N(n)^2, r_{2,P}(n)$ and $r_{2,P}(n)^2$. We also use these generating functions to obtain asymptotic estimates of the average values for all these functions.
Item Type: Article pubdom FALSE Dirichlet series, sum of squares, closed forms, binary quadratic forms, disjoint discriminants, L-functions 11-xx Number theory > 11Mxx Zeta and $L$-functions: analytic theory11-xx Number theory > 11Exx Forms and linear algebraic groups UNSPECIFIED Users 1 not found. 24 Nov 2003 12 Jan 2015 15:54 https://docserver.carma.newcastle.edu.au/id/eprint/142