Demystification of Taylor,Laurent coefficients of Lerch,Hurwitz Zeta functions and Dirichlet L-Function at Unity and Zero and their Bounds

TL;DR

This paper derives explicit formulas and bounds for derivatives of Hurwitz, Lerch, and Dirichlet L-functions at special points, providing elementary approximate functional equations and interrelations among these functions.

Contribution

It introduces simple, explicit expressions and bounds for higher derivatives of these zeta functions at critical points using elementary methods.

Findings

Explicit bounds for derivatives of Hurwitz and Dirichlet L-functions.

Elementary approximate functional equations in the critical strip.

Interrelations among Hurwitz, Lerch, and Dirichlet L-functions.

Abstract

Using elementary methods,we obtain simple,explicit expressions and bounds of higher order derivatives of Hurwitz zeta function and consequently those of Dirichlet L-function and also,of Lerch's Zeta function at unity (and at Zero too)and also obtain their interrelations.We also state elementary complete forms of approximate functional equations of Hurwitz zeta function and Dirichlet L-function in the critical strip.