Large Sums of Fourier Coefficients of Cusp Forms
Abstrak
Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} λ_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq x}λ_f(x)$. It is conjectured that $S(x,f)=o(x\log x)$ in the range $x\geq k^ε$. Lamzouri proved in arXiv:1703.10582 [math.NT] 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 $ε>(\log k)^{-\frac{1}{8}}$ and $1\leq T\leq (\log k)^{\frac{1}{200}}$, we have $S(x,f)\ll \frac{x\log x}{T}$ in the range $x\geq k^ε$ provided that $L(s,f)$ has no more than $ε^2\log k/5000$ zeros in the region $\left\{s\,:\, \Re(s)\geq \frac34, \, |\Im(s)-φ| \leq \frac14\right\}$ for every real number $φ$ with $|φ|\leq T$.
Topik & Kata Kunci
Penulis (4)
Claire Frechette
Mathilde Gerbelli-Gauthier
Alia Hamieh
Naomi Tanabe
Akses Cepat
- Tahun Terbit
- 2023
- Bahasa
- en
- Sumber Database
- arXiv
- Akses
- Open Access ✓