Does The Riemann Zeta Function Satisfy A Differential Equation?

Keywords

Euler-MacLauren summation formula; Infinite order differential equation; Primary; Riemann zeta function; Secondary

Abstract

In Hilbert's 1900 address at the International Congress of Mathematicians, it was stated that the Riemann zeta function is the solution of no algebraic ordinary differential equation on its region of analyticity. It is natural, then, to inquire as to whether ζ(. z) satisfies any non-algebraic differential equation. In the present paper, an elementary proof that ζ(. z) formally satisfies an infinite order linear differential equation with analytic coefficients, T[ζ. -. 1]. =. 1/(. z-. 1), is given. We also show that this infinite order differential operator T may be inverted, and through inversion of T we obtain a series representation for ζ(. z) which coincides exactly with the Euler-MacLauren summation formula for ζ(. z). Relations to certain known results and specific values of ζ(. z) are discussed.

Publication Date

2-1-2015

Publication Title

Journal of Number Theory

Volume

147

Number of Pages

778-788

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.jnt.2014.08.013

Socpus ID

84908374202 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84908374202

This document is currently not available here.

Share

COinS