Uniform bounds for norms of theta series and arithmetic applications

TL;DR

This paper establishes uniform bounds for the Petersson norm of theta series, leading to improved asymptotic formulas for quadratic form representations and specific integer representation results with prime divisor constraints.

Contribution

It provides new uniform bounds for theta series norms, enhancing understanding of quadratic form representations and their arithmetic properties.

Findings

Improved asymptotic formula for the number of quadratic form representations.

Every integer n ≠ 0,4,7 mod 8 can be represented as a sum of squares and a cube with limited prime divisors.

Uniform bounds for the Petersson norm of the cuspidal part of theta series.

Abstract

We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer $n \neq 0,4,7 \,(\operatorname{mod}8)$ is represented as $n= x_1^2 + x_2^2 + x_3^3$ for integers $x_1,x_2,x_3$ such that the product $x_1x_2x_3$ has at most 72 prime divisors.