TY - JOUR
T1 - Hermite Expansion of the Riemann Zeta Function
AU - Li , Bang-He
JO - Journal of Mathematical Study
VL - 4
SP - 319
EP - 324
PY - 2016
DA - 2016/12
SN - 49
DO - http://doi.org/10.4208/jms.v49n4.16.01
UR - https://global-sci.org/intro/article_detail/jms/10116.html
KW - Riemann zeta function, Hermite expansion, Schwartz distributions.
AB - <p style="text-align: justify;">Let $ζ(s)$ be the Riemann zeta function, $s=\sigma+it$. For $0 &lt; \sigma &lt; 1$, 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 $$ψ_n(t)=(2^nn!\sqrt{\pi})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$&nbsp; $H_n(t)$ 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 $σ &gt; 0.$ In the deduction,
it is crucial to regard the zeta function as Fourier transfomations of Schwartz&#39;
distributions.</p>