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The pair correlation of zeros of functions in the Selberg class. (English) Zbl 0929.11030

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.

MSC:

11M41 Other Dirichlet series and zeta functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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