In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Final

6q-ANALOGUEOFRIEMANNZETAFUNCTION

3.q-analogsofzetafunctions

Inthissection,weassumeq∈Rwith0<q<1.Nowweconsidertheq-extensionoftheEulerzetafunctionasfollows:

∞ns

nq( 1)ζq,E(s)=[2]q

n=1

[n+x]sq

.

Notethatζq,E(s,x)isananalyticcontinuationinwholecomplexs-plane.

By(14)andTheorem1,wehavethefollowingtheorem.Theorem4.Foranypositiveintegerk,wehave(15)

ζq,E( k,x)=Ek,q

( k,1)

(x,q).

Ford∈Nwithd≡1(mod2),letχbeDirichletcharacterwithconductord.By(13),thegeneralizedq-Eulernumbersattachedtoχcanbede nedas(16)

( m,1)Em,χ,q=

[2]q

d

).

Fors∈C,wede ne(17)

Lq,E(s,χ)=[2]q

∞ χ(n)( 1)nqsn

n=1

[2]qd

s

[d] q

a=1

d

χ(a)( 1)aqsaζqd,E(s,

a