2023/08/11 by Claire Frechette, Frechette, Claire, Mathilde Gerbelli-Gauthier +5
Mathematics · Social Sciences · #11F12 #11F30 (Primary) 11F11 #11M41 (Secondary) #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies
paper · pdf · doi:10.48550/arxiv.2308.06311
openalex publication_date 2023/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let N be a fixed positive integer, and let f∈ Sk(N) be a primitive cusp form given by the Fourier expansion f(z)=∑n=1∞ λf(n)n(k-1)/(2)e(nz). We consider the partial sum S(x,f)=∑n≤ xλf(x). It is conjectured that S(x,f)=o(xlog x) in the range x≥ kε. 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)-(1)/(8) and 1≤ T≤ (log k)(1)/(200), we have S(x,f)≪ (xlog x)/(T) in the range x≥ kε provided that L(s,f) has no more than ε2log k/5000 zeros in the region \s : \Re(s)≥ \frac34, |\Im(s)-ϕ| ≤ \frac14\ for every real number ϕ with |ϕ|≤ T.