The prime number theorem and Hypothesis H with lower-order terms

Abstract

Let π and π′ be unitary automorphic cuspidal representations of GLm(QA) and GLm′(QA), respectively, where at least one of π and π′ is self-contragredient. Using the prime number theorem for Rankin–Selberg L-functions, we compute a sharper version of Selberg orthogonality that contains certain lower-order terms which depend on special values of the Rankin–Selberg L-function attached to the pair (π,π′) and a sum related to Hypothesis H. In a case by case analysis when m,m′⩽4 and Hypothesis H is known to be true, we show how the constants involved in the lower-order terms can be expressed in terms of special values of Rankin–Selberg convolutions of symmetric- and/or exterior-power L-functions. In addition to showing that these constants give arithmetic information about the representations π and π′, we demonstrate how Hypothesis H can be used to give analytic continuation of the L-functions involved in the computation of the constants.