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## Bounds for a spectral exponential sum

Journal of London Mathematical Society. 2019. Vol. 99. No. 2. P. 249-272.
Balkanova O., Frolenkov D.

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 $T\geq X^{1/6+2\theta/3+\epsilon}$. 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.