Counting solutions of quadratic congruences in several variables revisited

Abstract

Let $N_k(n,r,\mathbf{a})$ denote the number of incongruent solutions of the quadratic congruence $a_1{x}_1^2+\dots +a_k{x}_k^2\equiv n$ (mod $r$), where $\mathbf{a}=(a_1,\dots ,a_k)\in {\mathbb{Z}}^k$, $n\in \mathbb{Z}$, $r\in \mathbb{N}$. We give short direct proofs for certain less known compact formulas on $N_k(n,r,\mathbf{a})$, valid for $r$ odd, which go back to the work of Minkowski, Bachmann and Cohen. We also deduce some other related identities and asymptotic formulas which do not seem to appear in the literature.

20 pages, revised, asymptotic formulas improved/added