(untitled) (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 distributions with variable t : ζ(σ+it)=^∞_n=0a_n(σ)ψ_n(t), where ψ_n(t)=(2^nn!π)^-1 ⁄ 2e^-t^22 H_n(t), H_n(t) is the Hermite polynomial, and a_n(σ)=2π(-1)^n+1ψ_n(i(1-σ))+(-i)^n2π^∞_m=11m^σψ_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.