We study the value distribution of the Epstein zeta function for and a random lattice of large dimension . For any fixed and , we prove that the random variable has a limit distribution, which we give explicitly (here is the volume of the -dimensional unit ball). More generally, for any fixed , we determine the limit distribution of the random function , . After compensating for the pole at , we even obtain a limit result on the whole interval , and as a special case we deduce the following strengthening of a result by Sarnak and Strömbergsson concerning the height function of the flat torus : the random variable has a limit distribution as , which we give explicitly. Finally, we discuss a question posed by Sarnak and Strömbergsson as to whether there exists a lattice for which has no zeros in .
"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