On prime numbers and quadratic forms represented by positive-definite, primitive quadratic forms

Abstract

In this note we show that every positive-definite, integral, primitive, n-ary quadratic form with $n\ge 2$ represents infinitely many prime numbers and infinitely many primitive, non-equivalent, m-ary quadratic forms for each $2\le m\le n-1$. We do so via an inductive argument which only requires to know the statement for $n=2$ (proved by H. Weber in 1882), and elementary linear algebra. The result on the representation of prime numbers by n-ary quadratic forms for arbitrary $n>2$ can be deduced from theorems already known, but the proof below is more direct and seems to be new in the literature. As an application we establish a non-vanishing result for Fourier-Jacobi coefficients of Siegel modular forms of any degree, level and Dirichlet character, subject to a condition on the conductor of the character.