• DOI:10.1215/00127094-1903389
• Corpus ID: 9748462

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

@inproceedings{Sodergren2011OnTV,
  title={On the value distribution of the Epstein zeta function in the critical strip},
  author={Anders Sodergren},
  year={2011}
}

• Anders Sodergren
• Published 13 May 2011
• Mathematics

We study the value distribution of the Epstein zeta function En(L, s) for 0 < s < n 2 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 V −2c n En(·, cn) has a limit distribution, which we give explicitly (here Vn is the volume of the ndimensional unit ball). More generally, for any fixed ε > 0 we determine the limit distribution of the random function c 7→ V −2c n En(·, cn), c ∈ [ 1 4 + ε, 1 2 − ε]. 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 [15] concerning the height function hn(L) of the flat torus R/L: The random variable n { hn(L)− (log(4π)− γ+1) } + log n 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 L ⊂ R for which En(L, s) has no zeros in (0,∞).