# On the location of the zero-free half-plane of a random Epstein zeta function

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## Abstract

In this note we study, for a random lattice *L* of large dimension *n*, the supremum of the real parts of the zeros of the Epstein zeta function \(E_n(L,s)\) and prove that this random variable scaled by \(n^{-1}\) has a limit distribution, which we give explicitly. This limit distribution is studied in some detail; in particular we give an explicit formula for its distribution function. Furthermore, we obtain a limit distribution for the frequency of zeros of \(E_n(L,s)\) in vertical strips contained in the half-plane \(\mathfrak {R}s>\frac{n}{2}\).

## Notes

### Acknowledgements

We are grateful to Daniel Fiorilli and Svante Janson for inspiring discussions and helpful remarks. We are also grateful to the referee for asking about the density of zeros; this inspired us to add Theorem 2 to the paper. The second author thanks the Institute for Advanced Study for providing excellent working conditions.

## Supplementary material

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