Gaps Between the Zeros of Epstein's Zeta-Functions on the Critical Line

Abstract

Abstract. It is proved that Epstein's zeta-function ζ Q (s) related to a positive definite integral binary quadratic form has a zero 1 2 + iγ with T ≤ γ ≤ T + T 5/11+ε for sufficiently large positive numbers T . This improves a classical result of H.

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

“…(A. 15) Next, taking the argument of both sides of (A.11), we get For c not too large, the formula (A.21) can be used to compute f (c) numerically to a decent precision. We have implemented this in [27, numdensity.mpl].…”

“…225-227]) that a positive proportion of the zeros do lie on the critical line. For related results, see also [15] and [19].…”

A note on the gaps between zeros of Epstein's zeta-functions on the critical line

“…Then for any fixed ε > 0 and T ≥ T (ε, Q), there is a zero 1/2 + iγ of the corresponding Epstein zeta-function ζ Q (s) with | γ − T |≤ T 3/7+ε . (1.2) In this paper we indicate those steps in [4] which enabled us to improve the result of Jutila and Srinivas mentioned earlier. Therefore, for technical details the readers are urged to refer to [4] and [3].…”

“…Sankaranarayanan in 1995 [9] showed that the same result holds true for intervals of the type [T, T + cT 1/2 log T ]. In 2005, Jutila and Srinivas [4] proved that the same is true for intervals of the type [T, T + cT 5/11+ε ], thus surpassing the classical barrier of 1/2 in exponent of T . In this note we improve this result further.…”

“…The zeta-function ζ Q 1 2 + i(T + u) appearing in the integrand is now replaced with an appropriate approximate formula. Such a formula was derived in [4], Lemma 1. We state this as Lemma 2.…”