Does the Riemann zeta function satisfy a differential equation?

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.