Fourier expansion of the Riemann zeta function and applications

JOURNAL OF NUMBER THEORY

卷:211

Article

Elaissaoui, Lahoucine¹  Guennoun, Zine El Abidine¹ 

[1] Mohammed V Univ Rabat, Fac Sci, Dept Math, Rabat, Morocco

关键词: Riemann zeta function;    Riemann hypothesis;    Lindelof hypothesis;    Stieltjes constants;    Fourier series;    Hardy space;   

DOI  :  10.1016/j.jnt.2019.09.025

来源: Elsevier

【 摘 要 】

Text. We study the distribution of values of the Riemann zeta function ((s) on vertical lines Rs+iR, by using the theory of Hilbert space. We show among other things, that, c(s) has a Fourier expansion in the half-plane R-s >= 1/2 and its Fourier coefficients are the binomial transform involving the Stieltjes constants. As an application, we show explicit computation of the Poisson integral associated with the logarithm of zeta(s) - s/(s - 1). Moreover, we discuss our results with respect to the Riemann and Lindelof hypotheses on the growth of the Fourier coefficients. Video. For a video summary of this paper, please visit https://youtu.be/wI5fIJMeqp4. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free