Counting solutions of quadratic congruences in several variables revisited

Let $N_k(n,r,\boldsymbol{a})$ denote the number of incongruent solutions of the quadratic congruence $a_1x_1^2+\ldots+a_kx_k^2\equiv n$ (mod $r$), where $\boldsymbol{a}=(a_1,\ldots,a_k)\in {\Bbb Z}^k$, $n\in {\Bbb Z}$, $r\in {\Bbb N}$. We give short direct proofs for certain less known compact formulas on $N_k(n,r,\boldsymbol{a})$, 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.Comment: 20 pages, revised, asymptotic formulas improved/adde