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We prove an exact formula for the second moment of Rankin–Selberg
is a fixed holomorphic cusp form and
is summed over automorphic forms of a given level
. The formula is a reciprocity relation that exchanges the twist parameter
and the level
. The method involves the Bruggeman–Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.
positive integers, to all
. There are poles of the function corresponding to zeros of the Riemann zeta function and the spectral parameters of Maass forms. The analytic properties of this function are rather delicate. It turns out that the spectral expansion of the zeta function converges only in a left half-plane, disjoint from the region of absolute convergence of the Dirichlet series, even though they both are analytic expressions of the same meromorphic function on the entire complex plane.
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