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