The typical size of Hecke eigenvalue sums is o(√x)
Abstract
Let ℋk denote the set of normalized holomorphic Hecke cusp forms of weight k for the full modular group $ \mathrm{SL}_2(\mathbb Z)$. For each f∈ℋk, let λf(n) denote the corresponding eigenvalue of the normalized Hecke operator Tn. We prove nontrivial upper bounds for ∑f ∈ ℋkωf |∑n ≤ x λf(n)|2q, where k is a positive even integer, 1leqslant xleqslant k, 0leqslant q leqslant 1 and $ ω_f = \frac{Γ(k-1)} {(4π)^{k-1}\langle f,f\rangle} = \frac{2π^2} {(k-1)L(1,\mathrm{Sym}^2 f)}. $ Our estimates match the conjecturally sharp upper bound for 1leqslant xleqslant k0.99. In particular, whenever both x and k/x tend to infinity with k, we obtain ∑f ∈ ℋkωf |∑n ≤ x λf(n)|=o(√x). We also prove upper bounds for low moments of sums of Hecke eigenvalues associated with Hecke--Maass cusp forms for SL2(ℤ). We study these sums through the probabilistic model proposed by Cogdell--Michel. We also determine the order of magnitude of low moments of this probabilistic model, which is equivalent to determining the order of magnitude of low moments of Hecke eigenvalue sums in the limit.
Related
- Low moments of Hecke eigenvalue sums
- Effective Hecke eigenvalue equidistribution over the Atkin--Lehner subspaces
- Metrics with positive Ricci eigenvalue sums on 𝕋k× 𝕊m
- Large values of quadratic Hecke L-functions with prime-related moduli of the Gaussian field
- Non-vanishing of central values of angular Hecke L-functions
BibTeX
@misc{xu2026typical,
title = {{The typical size of Hecke eigenvalue sums is $o(\sqrtx)$}},
author = {Max Wenqiang Xu and Junren Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14915},
eprint = {2609.14915},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14915},
}
RIS
TY - UNPB
TI - The typical size of Hecke eigenvalue sums is o(√x)
AU - Max Wenqiang Xu
AU - Junren Zheng
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14915
UR - https://arxiv.org/abs/2609.14915
AB - Let ℋ_k denote the set of normalized holomorphic Hecke cusp forms of weight k for the full modular group $ \mathrm{SL}_2(\mathbb Z)$. For each f∈ℋ_k, let λ_f(n) denote the corresponding eigenvalue of the normalized Hecke operator T_n. We prove nontrivial upper bounds for ∑_f ∈ ℋ_kω_f |∑_n ≤ x λ_f(n)|^2q, where k is a positive even integer, 1leqslant xleqslant k, 0leqslant q leqslant 1 and $ ω_f = \frac{Γ(k-1)} {(4π)^{k-1}\langle f,f\rangle} = \frac{2π^2} {(k-1)L(1,\mathrm{Sym}^2 f)}. $ Our estimates match the conjecturally sharp upper bound for 1leqslant xleqslant k^0.99. In particular, whenever both x and k/x tend to infinity with k, we obtain ∑_f ∈ ℋ_kω_f |∑_n ≤ x λ_f(n)|=o(√x). We also prove upper bounds for low moments of sums of Hecke eigenvalues associated with Hecke--Maass cusp forms for SL_2(ℤ). We study these sums through the probabilistic model proposed by Cogdell--Michel. We also determine the order of magnitude of low moments of this probabilistic model, which is equivalent to determining the order of magnitude of low moments of Hecke eigenvalue sums in the limit.
ER -