Hermite Expansion of the Riemann Zeta Function (Journal of Mathematical Study, 2021)
Let $ζ(s)$ be the Riemann zeta function, $s=σ+it$. For $0 < σ < 1$, we expand $ζ(s)$ as the following series convergent in the space of slowly increasing distributionswith variable $t$ : $$ζ(σ+it)=\sum\limits^∞_{n=0}a_n(σ)ψ_n(t),$$ where $$ψ_n(t)=(2^nn!\sqrt{π})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$ $H_n(t)$ is the Hermite polynomial, and $$a_n(σ)=2π(-1)^{n+1}ψ_n(i(1-σ))+(-i)^n\sqrt{2π}\sum\limits^∞_{m=1}\frac{1}{m^σ}ψ_n(1nm).$$ 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.