Large Sums of Fourier Coefficients of Cusp Forms
Claire Frechette1 ; Mathilde Gerbelli-Gauthier2 ; Alia Hamieh3 ; Naomi Tanabe4
1 Boston College, Department of Mathematics, Chestnut Hill, MA 02467, USA
2 University of Toronto, Department of Mathematics, Toronto, ON M5S 2E4, Canada
3 University of Northern British Columbia, Department of Mathematics and Statistics, Prince George, BC V2N 4Z9, Canada
4 Bowdoin College, Department of Mathematics, Brunswick, ME 04011, USA
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 171-188

Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum _{n=1}^{\infty } \lambda _f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum _{n\le x}\lambda _f(n)$. It is conjectured that $S(x,f)=o(x\log x)$ in the range $x\ge k^{\epsilon }$. Lamzouri proved in [8] that this is true under the assumption of the Generalized Riemann Hypothesis (GRH) for $L(s,f)$. In this paper, we prove that this conjecture holds under a weaker assumption than GRH. In particular, we prove that given $\epsilon >(\log k)^{-\frac{1}{8}}$ and $1\le T\le (\log k)^{\frac{1}{200}}$, we have $S(x,f)\ll \frac{x\log x}{T}$ in the range $x\ge k^{\epsilon }$ provided that $L(s,f)$ has no more than $\epsilon ^2\log k/5000$ zeros in the region $\left\lbrace s\,:\, \Re (s)\ge \frac{3}{4}, \, |\Im (s)-\phi | \le \frac{1}{4}\right\rbrace $ for every real number $\phi $ with $|\phi |\le T$.

Soit $N$ un entier positif et soit $f \in S_k(N)$ une forme cuspidale primitive admettant le développement en série de Fourier $f(z)= \sum _{n=1}^{\infty } \lambda _f(n)n^{\frac{k-1}{2}}e(nz)$. Nous étudions les sommes partielles $S(x,f)= \sum _{n\le x}\lambda _f(n)$, pour lesquelles il est conjecturé que $S(x,f)=o(x\log x)$ quand $x\ge k^{\epsilon }$. Dans [8], Lamzouri démontre cette conjecture en supposant que $L(s,f)$ satisfasse l’hypothèse de Riemann géneralisée. Dans cet article, nous démontrons la conjecture sous des hypothèses plus faibles. Plus précisément, nous démontrons qu’étant donnés $\epsilon >(\log k)^{-\frac{1}{8}}$ et $1\le T\le (\log k)^{\frac{1}{200}}$, on a $S(x,f)\ll \frac{x\log x}{T}$ quand $x\ge k^{\epsilon }$, pourvu que pour tout nombre réel $\phi $ tel que $|\phi |\le T$, la fonction $L(s,f)$ n’ait pas plus de $\epsilon ^2\log k/5000$ zéros dans la région $\left\lbrace s\,:\, \Re (s)\ge \frac{3}{4}, \, |\Im (s)-\phi | \le \frac{1}{4}\right\rbrace $.

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DOI : 10.5802/jtnb.1318
Classification : 11F30, 11F11, 11F12, 11M41
Keywords: Modular forms, Sums of Fourier coefficients, zeros of $L$-functions, mean values of multiplicative functions.

Claire Frechette 1 ; Mathilde Gerbelli-Gauthier 2 ; Alia Hamieh 3 ; Naomi Tanabe 4

1 Boston College, Department of Mathematics, Chestnut Hill, MA 02467, USA
2 University of Toronto, Department of Mathematics, Toronto, ON M5S 2E4, Canada
3 University of Northern British Columbia, Department of Mathematics and Statistics, Prince George, BC V2N 4Z9, Canada
4 Bowdoin College, Department of Mathematics, Brunswick, ME 04011, USA
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Claire Frechette; Mathilde Gerbelli-Gauthier; Alia Hamieh; Naomi Tanabe. Large Sums of Fourier Coefficients of Cusp Forms. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 171-188. doi: 10.5802/jtnb.1318

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