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arXiv · 2607.10786

Sums of Kloosterman sums formed with modular symbols

Nikolaos Diamantis·Solomon Friedberg·Fredrik Strömberg
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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 $\Re(s)>1/2$, 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.

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Nikolaos Diamantis, Solomon Friedberg, Fredrik Strömberg. 2026-07-12. Sums of Kloosterman sums formed with modular symbols. https://arxiv.org/abs/2607.10786

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