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

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

2011-05-13

arxiv.org/abs/1105.2847v1

Mathematics

Number Theory

36 pages, 2 figures

Scientific paper

We study the value distribution of the Epstein zeta function $E_n(L,s)$ for $00$ we determine the limit distribution of the random function $c\mapsto V_n^{-2c}E_n(\cdot,cn)$, $c\in[1/4 +\ve, 1/2-\ve]$. After compensating for the pole at $c=\frac12$ we even obtain a limit result on the whole interval $[\frac14+\ve,\frac12]$, and as a special case we deduce the following strengthening of a result by Sarnak and Str\"ombergsson concerning the height function $h_n(L)$ of the flat torus $\R^n/L$: The random variable $n\big\{h_n(L)-(\log(4\pi)-\gamma+1)\big\}+\log n$ has a limit distribution as $n\to\infty$, which we give explicitly. Finally we discuss a question posed by Sarnak and Str\"ombergsson as to whether there exists a lattice $L\subset\R^n$ for which $E_n(L,s)$ has no zeros in $(0,\infty)$.

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