Uniform bounds for norms of theta series and arithmetic applications

Fabian Waibel

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arXiv · 2012.04311

Uniform bounds for norms of theta series and arithmetic applications

Fabian Waibel

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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.

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BibTeX RIS

Fabian Waibel. 2020-12-08. Uniform bounds for norms of theta series and arithmetic applications. https://arxiv.org/abs/2012.04311

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