Infinite order linear differential equation satisfied by p-adic Hurwitz-type Euler zeta functions

作者：Hu, Su; Kim, Min-Soo^*

来源：ABHANDLUNGEN AUS DEM MATHEMATISCHEN SEMINAR DER UNIVERSITAT HAMBURG, 2021, 91(1): 117-135.

DOI：10.1007/s12188-021-00234-2

• p-adic Hurwitz-type Euler zeta function
• Differential equation

In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function zeta(s) is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder (J Number Theory 147:778-788, 2015) considered the question of whether zeta(s) satisfies a non-algebraic differential equation and showed that it formally satisfies an infinite order linear differential equation. Recently, Prado and Klinger-Logan (J Number Theory 217:422-442, 2020) extended Van Gorder's result to show that the Hurwitz zeta function zeta(s, a) is also formally satisfies a similar differential equation @@@ T[zeta(s, a) - 1/a(s)] = 1/(s - 1)a(s-1). @@@ But unfortunately in the same paper they proved that the operator T applied to Hurwitz zeta function zeta(s, a) does not converge at any point in the complex plane C. In this paper, by defining T-p(a), a p-adic analogue of Van Gorder's operator T, we establish an analogue of Prado and Klinger-Logan's differential equation satisfied by zeta(p,E)(s, a) which is the p-adic analogue of the Hurwitz-type Euler zeta functions @@@ zeta E(s, a) = Sigma(infinity )(n=0)(-1)(n)/(n + a)(s). @@@ In contrast with the complex case, due to the non-archimedean property, the operator T-p(a) applied to the p-adic Hurwitz-type Euler zeta function zeta(p,E)(s, a) is convergent p-adically in the area of s is an element of Z(p) with s not equal 1 and a is an element of K with vertical bar a vertical bar p > 1, where K is any finite extension of Q(p) with ramification index over Q(p) less than p - 1.