Year: 2016
Author: Bang-He Li
Journal of Mathematical Study, Vol. 49 (2016), Iss. 4 : pp. 319–324
Abstract
Let $ζ(s)$ be the Riemann zeta function, $s=\sigma+it$. For $0 < \sigma < 1$, we expand $ζ(s)$ as the following series convergent in the space of slowly increasing distributions with variable $t$ : $$ζ(\sigma+it)=\sum\limits^∞_{n=0}a_n(\sigma)ψ_n(t),$$ where $$ψ_n(t)=(2^nn!\sqrt{\pi})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$ $H_n(t)$ is the Hermite polynomial, and $$a_n(σ)=2\pi(-1)^{n+1}ψ_n(i(1-σ))+(-i)^n\sqrt{2\pi}\sum\limits^∞_{m=1}\frac{1}{m^σ}ψ_n(1nm).$$ This paper is concerned with the convergence of the above series for $σ > 0.$ In the deduction, it is crucial to regard the zeta function as Fourier transfomations of Schwartz' distributions.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v49n4.16.01
Journal of Mathematical Study, Vol. 49 (2016), Iss. 4 : pp. 319–324
Published online: 2016-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 6
Keywords: Riemann zeta function Hermite expansion Schwartz distributions.