Extreme values for Sn(σ, t) near the critical line

JOURNAL OF NUMBER THEORY

卷:200

Article

Chirre, Andres¹ 

[1] IMPA Inst Nacl Matemat Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, RJ, Brazil

关键词: Riemann zeta function;    Riemann hypothesis;    Argument;   

DOI  :  10.1016/j.jnt.2018.12.009

来源: Elsevier

【 摘 要 】

Let S(sigma, t) = 1/pi arg zeta(sigma + it) be the argument of the Riemann zeta function at the point sigma + it of the critical strip. For n >= 1 and t > 0 we define S-n (sigma ,t) = integral(t)(0) Sn-1 (sigma, tau) d tau + delta(n,sigma), where delta(n,sigma) is a specific constant depending on sigma and n. Let 0 <= beta < 1 be a fixed real number. Assuming the Riemann hypothesis, we show lower bounds for the maximum of the function S-n (sigma, t) on the interval T-beta <= t <= T and near to the critical line, when n equivalent to 1 mod 4. Similar estimates are obtained for vertical bar S-n (sigma, t)vertical bar when n not equivalent to 1 mod 4. This extends the results of Bondarenko and Seip [7] for a region near the critical line. In particular we obtain some omega results for these functions on the critical line. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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