Explicit Formula for the Mean Square of Dirichlet L-Functions to Prime Power Moduli
Individual values of L-functions inside the critical strip have great arithmetic significance (such as the Riemann hypothesis) but are presently not well-understood. In several important problems in analytic number theory, including how primes are distributed, estimates such as power moments provide critical input about L-functions. We derive an explicit formula for the second moment in the family of Dirichlet L- functions to a prime power modulus. Motohashi derived explicit formulas for the second and fourth moments of the Riemann zeta function along the critical line, and our formula provides the analogue for the second moment of Dirichlet L-functions. There are many estimates on the power moments for Dirichlet L-functions, but the present work does not include an error term. Our main theorem expresses the second moment in the family of Dirichlet L-functions to a prime power modulus as a sum consisting of five summands. One of the summands consists of contributions from diagonal and near-diagonal terms. The remaining four o-diagonal terms arise from residues and integrals from contour shifts. Two of these summands are sums weighted by the divisor function and certain transforms of the weight function. We obtain asymptotics with a power saving error term when averaging over a family that is of size above the square root of the conductor. We employ a variety of tools: orthogonality of characters, discrete and continuous Fourier and Mellin transforms, analytic continuation, contour shifting, functional equations, and p-adic stationary phase in asymptotics.
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