It is a classical problem to consider the asymptotic behavior of the Fourier coefficients of modular forms and their hidden structures. One of the common approaches is to study the asymptotic of their summatory functions. In this talk, we establish an upper bound for the sum $\sum_{n\leq x} \lambda_f(n)$ of the Fourier coefficients of primitive cusp forms for $x\geq k^\epsilon$ under a weaker assumption than GRH, motivated by the work of Granville and Soundararajan, where they study the case of Dirichlet characters. We also demonstrate a concrete connection between large coefficient sums and zeros of $L(s, f)$. This project originated from the Women In Numbers 6 Research Workshop and is worked jointly with Claire Frechette, Mathilde Gerbelli-Gauthier, and Alia Hamieh.
Large Sums of Fourier Coefficients of Cusp Forms
Naomi Tanabe, Bowdoin College
Authors: Claire Frechette, Mathilde Gerbelli-Gauthier, Alia Hamieh, Naomi Tanabe
2023 AWM Research Symposium
Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]