Zeros of the Epstein zeta function to the right of the critical line

• Published in: Mathematical proceedings of the Cambridge Philosophical Society Vol. 171; no. 2; pp. 265 - 276
• Main Author: LAMZOURI, YOUNESS
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2021; Cambridge University Press (CUP)
• Subjects: Fields (mathematics); Hypotheses; Mathematics; Number Theory; Quadratic forms; Real numbers; 11M41; 11E45; Epstein zeta function; relations with automorphic forms and fucntions; other Dirichlet series and Zeta functions
• ISSN: 0305-0041; 1469-8064
• DOI: 10.1017/S0305004120000213

Let E(s, Q) be the Epstein zeta function attached to a positive definite quadratic form of discriminant D < 0, such that h(D) ≥ 2, where h(D) is the class number of the imaginary quadratic field ${\mathbb{Q}} (\sqrt D)$ . We denote by ${N_E}({\sigma _1},{\sigma _2},T)$ the number of zeros of $[E(s,Q)$ in the rectangle ${\sigma _1} < {\mathop{\rm Re}\nolimits} (s) \le {\sigma _2}$ and $T \le {\mathop{\rm Im}\nolimits} (s) \le 2T$ , where $1/2 < {\sigma _1} < {\sigma _2} < 1$ are fixed real numbers. In this paper, we improve the asymptotic formula of Gonek and Lee for ${N_E}({\sigma _1},{\sigma _2},T)$ , obtaining a saving of a power of log T in the error term.