Uniform bounds for norms of theta series and arithmetic applications

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 $n \neq 0,4,7 \,(\textrm{mod}\ 8)$ 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.

Finiteness theorems for universal sums of squares of almost primes

“…where b Q,δ,j (n) is the coefficient of the theta function Θ Q at the corresponding cusp. Trivially bounding det(D) ≤ 2 ℓ ∆ Q for the diagonal form defined before [23,Lemma 11], [23, Lemma 12] implies that…”