PreprintCC BY 4.0arXiv
Sums of Kloosterman sums formed with modular symbols
Abstract
We study sums of Kloosterman sums formed with a modular symbol. Employing Tauberian methods, we first give an estimate for a (Riesz) sum of Ramanujan sums formed with a modular symbol. We further define a zeta function that is analogous to the Selberg zeta function, establish its continuation to ℜ(s)>1/2, give estimates for its growth and use this to prove a cancellation statement for sums of these twisted Kloosterman sums. We explain the connection of this construction to the eigenvalue 1/4 problem and formulate an analogue of Linnik's conjecture. Finally, we present numerical evidence that there is cancellation and also that the Kloosterman sums with a modular symbol are not correlated with classical Kloosterman sums.
Related
- Kloosterman Sums and Plücker Height Zeta Functions
- Bessel Distributions and Kloosterman Sums
- Galois representations and modularity for twisted moments of Kloosterman sums
- Stevens' Approach to Kloosterman Sums on Split Groups
- Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors
BibTeX
@misc{diamantis2026sums,
title = {{Sums of Kloosterman sums formed with modular symbols}},
author = {Nikolaos Diamantis and Solomon Friedberg and Fredrik Strömberg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.10786},
eprint = {2607.10786},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10786},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sums of Kloosterman sums formed with modular symbols
AU - Nikolaos Diamantis
AU - Solomon Friedberg
AU - Fredrik Strömberg
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2607.10786
UR - https://arxiv.org/abs/2607.10786
AB - We study sums of Kloosterman sums formed with a modular symbol. Employing Tauberian methods, we first give an estimate for a (Riesz) sum of Ramanujan sums formed with a modular symbol. We further define a zeta function that is analogous to the Selberg zeta function, establish its continuation to ℜ(s)>1/2, give estimates for its growth and use this to prove a cancellation statement for sums of these twisted Kloosterman sums. We explain the connection of this construction to the eigenvalue 1/4 problem and formulate an analogue of Linnik's conjecture. Finally, we present numerical evidence that there is cancellation and also that the Kloosterman sums with a modular symbol are not correlated with classical Kloosterman sums.
ER -