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In many different settings, it is possible to determine statistics about spacings (pair correlation, small gaps, large gaps, champions, etc.), for instance

  • prime numbers
  • Laplacian eigenvalues on a lattice
  • zeros of L-functions

In the latter setting, there are an expansive literature about correlations between zeros of L-functions, but what is known or attempted concerning the other problems (I have in mind essentially small and large gaps)?

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Large gaps: there is the striking result that the gap size is bounded, see The largest gap between zeros of entire L-functions is less than 41.54 (2017).

Small gaps: according to arXiv:1202.2671 there exist many Dirichlet L-functions that have a pair of consecutive zeros closer together than 0.37 times their average spacing.

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Typing "gaps between zeros of Selberg class" on Google gives as the fifth research result a survey by Caroline Turnage-Butterbaugh that may be helpful.

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