Euler Numbers and Polynomials Associated with Zeta Functions

For s ∈ ℂ, the Euler zeta function and the Hurwitz‐type Euler zeta function are defined by ζE(s)=2∑n=1∞((−1)n/ns), and ζE(s,x)=2∑n=0∞((−1)n/(n+x)s). Thus, we note that the Euler zeta functions are entire functions in whole complex s‐plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is, ζE(−k)=Ek∗, and ζE(−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.