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Abstract. For any two functions F and G in the Selberg class we prove explicit formulas which relate sums over pairs of zeros, of the form: f rF rG ÀTgF Y gG T to sums over prime powers, of the form: T vF nvG ngn p n2 where f and g are test functions such that f is the Mellin transform of g. As a consequence we ®nd that the Weak Pair Correlation Conjecture for functions in the Selberg class is essentially equivalent to the Selberg Orthonormality Conjectures. 1991 Mathematics Subject Classi®cation: 11M41. 1 Introduction In 1989, Selberg [11] de®ned a general class S of Dirichlet series that admit analytic continuation, functional equation and an Euler product. Presumably, this class includes all the automorphic L-functions, but this has not been established since we do not yet know the Ramanujan conjecture for GLn for n 2. Maybe S even coincides with the class of automorphic L-functions in GLn . The class S consists of Dirichlet series F s y aF nan s satisfying the foln1 lowing axioms: (i) there exists an integer m 0 such that s À 1 m F s is an entire function of ®nite order;
Forum Mathematicum – de Gruyter
Published: Jan 29, 2002
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