28 research outputs found
Prime geodesic theorem for the Picard manifold
Let be the Picard group and be the
three-dimensional hyperbolic space. We study the Prime Geodesic Theorem for the
quotient , called the Picard manifold, obtaining an error
term of size , where denotes a
subconvexity exponent for quadratic Dirichlet -functions defined over
Gaussian integers.Comment: Corrected subconvexity estimate in (3.24
Convolution formula for the sums of generalized Dirichlet L-functions
Using the Kuznetsov trace formula, we prove a spectral decomposition for the
sums of generalized Dirichlet -functions. Among applications are an explicit
formula relating norms of prime geodesics to moments of symmetric square
-functions and an asymptotic expansion for the average of central values of
generalized Dirichlet -functions.Comment: to appear in Revista Matem\'atica Iberoamerican
An explicit formula for the second moment of Maass form symmetric square L-functions
Altres ajuts: Olga Balkanova's research was funded by RFBR, project number 19-31-60029. Dmitry Frolenkov's research was supported by the Theoretical Physics and Mathematics Advancement Foundation "BASIS".We prove an explicit formula for the second moment of symmetric square L-functions associated to Maass forms for the full modular group. In particular, we show how to express the considered second moment in terms of dual second moments of symmetric square L-functions associated to Maass cusp forms of levels 4, 16, and 64
Bounds for a spectral exponential sum
We prove new upper bounds for a spectral exponential sum by refining the
process by which one evaluates mean values of -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 .
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.Comment: final version, to appear in J. Lond. Math. So