Weighted quadratic partitions modulo $P^m$- a new formula and a new demonstration

Abstract

Let $Q({\bf{x}}) = Q(x_1 ,x_2 ,...,x_n )$ be a quadratic form over $\mathbb{Z}$, $p$ be an odd prime. Let $V = V_Q = V_{p^m } $ denote the set of zeros of $Q({\mathbf{x}})$ in $\mathbb{Z}_{p^m }$ and $\left| V \right|$ denotes the cardinality of $V$. Set $ \phi (V_{p^m } ,{\mathbf{y}}) = \sum _{{\mathbf{x}} \in V} e_{p^m } ({\mathbf{x}} \cdot {\mathbf{y}})$ for ${\mathbf{y}} \ne {\mathbf{0}}$ and $\phi (V_{p^m } ,{\mathbf{y}}) = \left| {V_{p^m } } \right| - p^{m(n - 1)}$ for ${\mathbf{y}} = {\mathbf{0}}.$ In this paper we shall give a formula for the calculation of the function $\phi (V,{\mathbf{y}}).$