Quadratic minima and modular forms II

TL;DR

This paper establishes upper bounds on the gaps in Fourier coefficients of level two modular forms and on integers represented by certain quadratic forms, extending previous partial results.

Contribution

It provides new upper bounds for Fourier coefficient gaps and represented integers in level two quadratic forms, advancing the results from part I.

Findings

Upper bounds on Fourier coefficient gaps

Upper bounds on integers represented by quadratic forms

Extension of previous partial results

Abstract

We give upper bounds on the size of the gap between a non-zero constant term and the next non-zero Fourier coefficient of an entire level two modular form. We give upper bounds for the minimum positive integer represented by a level two even positive-definite quadratic form. These bounds extend partial results in part I.