On the representation numbers satisfying partially multiplicative relations

Abstract

Let rQ(n) be the representation number of a nonnegative integer n by an integral positive definite quadratic form Q in 2k variables. Let N be the level of Q. For a fixed prime p∤N, we assume that rQ(n) satisfies the relation rQ(1)rQ(p2n)=rQ(p2)rQ(n) for all positive integers n with p∤n. We actually show that this relation holds for any prime p∤N when (−1)kN is a fundamental discriminant.