Revista Matemática Iberoamericana


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Volume 35, Issue 7, 2019, pp. 1973–1995
DOI: 10.4171/rmi/1106

Published online: 2019-07-12

Convolution formula for the sums of generalized Dirichlet $L$-functions

Olga Balkanova[1] and Dmitry Frolenkov[2]

(1) Chalmers University of Technology, Göteborg, Sweden
(2) Khabarovsk Division of the Institute for Applied Mathematics and Steklov Mathematical Institute, Moscow, Russian Federat

Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet $L$-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square $L$-functions and an asymptotic expansion for the average of central values of generalized Dirichlet $L$-functions.

Keywords: Generalized Dirichlet $L$-functions, prime geodesic theorem, Kuznetsov trace formula, generalized Kloosterman sums, symmetric square $L$-functions

Balkanova Olga, Frolenkov Dmitry: Convolution formula for the sums of generalized Dirichlet $L$-functions. Rev. Mat. Iberoam. 35 (2019), 1973-1995. doi: 10.4171/rmi/1106