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15 January 2013 On the value distribution of the Epstein zeta function in the critical strip
Anders Södergren
Duke Math. J. 162(1): 1-48 (15 January 2013). DOI: 10.1215/00127094-1903389

Abstract

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,).

Citation

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Anders Södergren. "On the value distribution of the Epstein zeta function in the critical strip." Duke Math. J. 162 (1) 1 - 48, 15 January 2013. https://doi.org/10.1215/00127094-1903389

Information

Published: 15 January 2013
First available in Project Euclid: 14 January 2013

zbMATH: 1285.11065
MathSciNet: MR3011871
Digital Object Identifier: 10.1215/00127094-1903389

Subjects:
Primary: 11E45
Secondary: 11P21, 60G55

Rights: Copyright © 2013 Duke University Press

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Vol.162 • No. 1 • 15 January 2013
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