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On The Value Distribution Of The Epstein Zeta Function In The Critical Strip
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2013 (English)In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 162, no 1, p. 1-48Article in journal (Refereed) Published
##### Abstract [en]

We study the value distribution of the Epstein zeta function E-n (L, s) for 0 <s < n/2 and a random lattice L of large dimension n. For any fixed c is an element of (1/4, 1/2) and n -> infinity, we prove that the random variable V-n(-2c) E-n(.,cn) has a limit distribution, which we give explicitly (here V-n is the volume of the n-dimensional unit ball). More generally, for any fixed epsilon > 0, we determine the limit distribution of the random function c bar right arrow V-n(-2c) E-n(., cn), c epsilon [1/4 + epsilon, 1/2 - epsilon]. After compensating for the pole at c = 1/2, we even obtain a limit result on the whole interval [1/4 + epsilon, 1/2], and as a special case we deduce the following strengthening of a result by Sarnak and Strombergsson concerning the height function h(n) (L) of the flat torus R-n/L: the random variable n{h(n) (L) - (log(4 pi) - gamma + 1)} + log n has a limit distribution as n -> infinity, which we give explicitly. Finally, we discuss a question posed by Sarnak and Strombergsson as to whether there exists a lattice L subset of R-n for which E-n(L, s) has no zeros in (0, infinity).

##### Place, publisher, year, edition, pages
2013. Vol. 162, no 1, p. 1-48
##### National Category
Natural Sciences Mathematics
##### Identifiers
ISI: 000314079400001OAI: oai:DiVA.org:uu-195383DiVA, id: diva2:607761
Available from: 2013-02-25 Created: 2013-02-25 Last updated: 2017-12-06Bibliographically approved

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##### By organisation
Analysis and Applied Mathematics
##### In the same journal
Duke mathematical journal
##### On the subject
Natural SciencesMathematics

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Cite
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