Explicit formulas for the pair correlation of zeros of functions in the Selberg class

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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: Σ-T ≤ γF, γG ≤ T f(ρF + ρG) to sums over prime powers, of the form: T/πΣn ≥ 2 ΛF(n)ΛG(n)g(n) where f and g are test functions such that f is the Mellin transform of g. As a consequence we find that the Weak Pair Correlation Conjecture for functions in the Selberg class is essentially equivalent to the Selberg Orthonormality Conjectures.

Original languageEnglish (US)
Pages (from-to)65-83
Number of pages19
JournalForum Mathematicum
Volume14
Issue number1
DOIs
StatePublished - 2002
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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