Duke Mathematical Journal

On the value distribution of the Epstein zeta function in the critical strip

Anders Södergren

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We study the value distribution of the Epstein zeta function En(L,s) for 0<s<n2 and a random lattice L of large dimension n. For any fixed c(1/4,1/2) and n, we prove that the random variable Vn2cEn(,cn) has a limit distribution, which we give explicitly (here Vn is the volume of the n-dimensional unit ball). More generally, for any fixed ε>0, we determine the limit distribution of the random function cVn2cEn(,cn), c[1/4+ε,1/2ε]. After compensating for the pole at c=1/2, we even obtain a limit result on the whole interval [1/4+ε,1/2], and as a special case we deduce the following strengthening of a result by Sarnak and Strömbergsson concerning the height function hn(L) of the flat torus Rn/L: the random variable n{hn(L)(log(4π)γ+1)}+logn has a limit distribution as n, which we give explicitly. Finally, we discuss a question posed by Sarnak and Strömbergsson as to whether there exists a lattice LRn for which En(L,s) has no zeros in (0,).

Article information

Duke Math. J., Volume 162, Number 1 (2013), 1-48.

First available in Project Euclid: 14 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E45: Analytic theory (Epstein zeta functions; relations with automorphic
Secondary: 11P21: Lattice points in specified regions 60G55: Point processes


Södergren, Anders. On the value distribution of the Epstein zeta function in the critical strip. Duke Math. J. 162 (2013), no. 1, 1--48. doi:10.1215/00127094-1903389. https://projecteuclid.org/euclid.dmj/1358172073

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