Title

Does the Riemann zeta function satisfy a differential equation?

Authors

Authors

R. A. Van Gorder

Comments

Authors: contact us about adding a copy of your work at STARS@ucf.edu

Abbreviated Journal Title

J. Number Theory

Keywords

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

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 zeta(z) satisfies any non-algebraic differential equation. In the present paper, an elementary proof that zeta(z) formally satisfies an infinite order linear differential equation with analytic coefficients, T[zeta - 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 zeta(z) which coincides exactly with the Euler-MacLauren summation formula for zeta(z). Relations to certain known results and specific values of zeta(z) are discussed. (C) 2014 Elsevier Inc. All rights reserved.

Journal Title

Journal of Number Theory

Volume

147

Publication Date

1-1-2015

Document Type

Article

Language

English

First Page

778

Last Page

788

WOS Identifier

WOS:000345194100044

ISSN

0022-314X

Share

COinS