Murty, M. Ram; Perelli, Alberto The pair correlation of zeros of functions in the Selberg class. (English) Zbl 0929.11030 Int. Math. Res. Not. 1999, No. 10, 531-545 (1999). An interesting class of Dirichlet series \({\mathcal S}\) with properties typical of the familiar \(L\)-functions, has been introduced by A. Selberg [Collected papers, Volume II, 47-63 (Springer 1991; Zbl 0729.11001), see also Zbl 0787.11037], where a variety of conjectures about \({\mathcal S}\), including an analogue of the Riemann hypothesis, are made.In this paper a pair correlation function \({\mathcal F}_{F,G} (\alpha,T)\), where \(F,G\in {\mathcal S}\), is defined. Assuming the relevant Riemann hypotheses, an asymptotic formula for \({\mathcal F}_{F,G} (\alpha,T)\) is obtained, for \(|\alpha |<1/d_F\), where \(d_F\) is the ‘degree’ of \(F(s)\). (Note that \({\mathcal F}_{F,G}(\alpha,T)\) is not symmetric between \(F\) and \(G.)\) Unfortunately the asymptotic formula depends on a hypothesis about the behaviour of \(\sum_{n\leq x}\Lambda_F (n) \Lambda_G(n)\). The authors make the natural analogue of H. L. Montgomery’s pair correlation conjecture, [Analytic Number Theory, Proc. Sympos. Pure Math. 24, St. Louis Univ. Missouri 1972, 181-193 (1973; Zbl 0268.10023)]. From this they deduce that ‘almost all’ zeros of a primitive function \(F(s)\) are simple, and that ‘almost all’ zeros of distinct primitive functions \(F(s)\) and \(G(s)\) are different. Moreover, assuming that the asymptotic formula holds even for a single non-zero \(\alpha\), they are able to show that factorization into primitive elements of \({\mathcal S}\) is unique. Reviewer: D.R.Heath-Brown (Oxford) Cited in 1 ReviewCited in 16 Documents MSC: 11M41 Other Dirichlet series and zeta functions 11M06 \(\zeta (s)\) and \(L(s, \chi)\) Keywords:Selberg conjectures; Selberg class; Dirichlet series; Riemann hypothesis; pair correlation; asymptotic formula Citations:Zbl 0787.11037; Zbl 0729.11001; Zbl 0268.10023 PDFBibTeX XMLCite \textit{M. R. Murty} and \textit{A. Perelli}, Int. Math. Res. Not. 1999, No. 10, 531--545 (1999; Zbl 0929.11030) Full Text: DOI