Bounds for a spectral exponential sum

Olga Balkanova, Dmitry Frolenkov

Type: Article

Publication Date: 2018-09-11

Citations: 15

DOI: https://doi.org/10.1112/jlms.12174

Abstract

We prove new upper bounds for a spectral exponential sum by refining the process by which one evaluates mean values of L-functions multiplied by an oscillating function. In particular, we introduce a method which is capable of taking into consideration the oscillatory behaviour of the function. This gives an improvement of the result of Luo and Sarnak when T ⩾ X 1 / 6 + 2 θ / 3 + ε . Furthermore, this proves the conjecture of Petridis and Risager in some ranges. Finally, this allows obtaining a new proof of the Soundararajan–Young error estimate in the prime geodesic theorem.

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