On the location of the zero-free half-plane of a random Epstein zeta function

Abstract

Abstract. In this note we study, for a random lattice L of large dimension n, the supremum of the real parts of the zeros of the Epstein zeta function En(L, s) and prove that this random variable scaled by n −1 has a limit distribution, which we give explicitly. This limit distribution is studied in some detail; in particular we give an explicit formula for its distribution function. Furthermore, we obtain a limit distribution for the frequency of zeros of En(L, s) in vertical strips contained in the half-plane ℜs > n 2 .

On the universality of the Epstein zeta function

“…This corollary complements the recent results in [19], that treats zeros in the halfplane Re(s) > 1, by showing that also the zeros in the strip {s : 3 4 < Re(s) < 1} violates the Riemann hypothesis in a strong way. We also note that the corresponding result for c-values (values s such that E n L, ns 2 = c) follows by a similar argument.…”

“…However, we stress that there are also important differences between E n (L, s) and ζ(s). Typically E n (L, s) has no Euler product and it is well known that the Riemann hypothesis for E n (L, s) generally fails (cf., e.g., [19] and the references therein). 3 Let V n denote the volume of the n-dimensional unit ball.…”

Adelic Rogers integral formula

“…The quoted problem above, in particular, had been intensively studied in a series of works by Rogers (see, e.g., [27,28]) and Schmidt (e.g., [33,34]; also see the references in [34]). Recently, their ideas and methods have been further developed by the author ( [15,16]), Södergren (e.g., [41,42]), and Strömbergsson-Södergren [40], with applications to the study of the Epstein zeta functions ( [39,43]).…”

Adelic Rogers integral formula

“…Recently, their ideas and methods have been further developed by the author ( [12], [13]), Södergren (e.g. [37], [38]) and ), with applications to the study of the Epstein zeta functions ( [39], [35]).…”