Euler Numbers and Polynomials Associated with Zeta Functions

Taekyun Kim

doi:10.1155/2008/581582

Abstract

Fors∈ℂ, the Euler zeta function and the Hurwitz-type Euler zeta function are defined byζE(s)=2∑n=1∞((−1)n/ns), andζE(s,x)=2∑n=0∞((−1)n/(n+x)s). Thus, we note that the Euler zeta functions are entire functions in whole complexs-plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is,ζE(−k)=Ek∗, andζE(−k,x)=Ek∗(x). We give some interesting identities between the Euler numbers and the zeta functions. Finally, we will give the new values of the Euler zeta function at positive even integers.

Highlights

Throughout this paper, Z, Q, C, Zp, Qp, and Cp will, respectively, denote the ring of rational integers, the field of rational numbers, the field of complex numbers, the ring p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp
Riemann did develop the theory of analytic continuation needed to rigorously define ζ s for all s ∈ C − {0}
1.13 he derived an explicit formula for the prime numbers in terms of zeros of the zeta function

Summary

Recommended by Lance Littlejohn

For s ∈ C, the Euler zeta function and the Hurwitz-type Euler zeta function are defined by ζE s. We note that the Euler zeta functions are entire functions in whole complex s-plane, and these zeta functions have the values and ζE of the −k, x. Euler numbers or the Euler polynomials at Ek∗ x. We give some interesting identities negative between integers. We will give the new values of the Euler zeta function at positive even integers. Copyright q 2008 Taekyun Kim. Copyright q 2008 Taekyun Kim

Introduction

Zp lim

It is easy to see that z cot z