Some relations involving the higher derivatives of the Riemann zeta function

TL;DR

This paper derives new integral representations for the higher derivatives of the Riemann zeta function, connecting them with the digamma function, Stieltjes constants, and Lehmer constants, advancing understanding of these special functions.

Contribution

It introduces novel formulas expressing derivatives of the Riemann zeta function in terms of integrals involving special constants, enhancing analytical tools for studying zeta function derivatives.

Findings

Derived integral expressions involving the digamma function for higher derivatives of zeta

Presented formulas relating derivatives of zeta to Stieltjes constants

Expressed derivatives of zeta in terms of Lehmer constants

Abstract

We show that the higher derivatives of the Riemann zeta function may be expressed in terms of integrals involving the digamma function. Related integrals for the Stieltjes constants are also shown. We also present a formula for the derivatives of the Riemann zeta function entirely in terms of the Lehmer constants.