# Euler sums on arithmetic progressions

### New Zealand Journal of Mathematics

Vol. 39, (2009), Pages 1-18

Minking Eie

Department of Mathematics

National Chung Cheng University

Minhsiung, Chiayi 62145

Taiwan

Wen-Chin Liaw

Department of Mathematics

National Chung Cheng University

Minhsiung, Chiayi 62145

Taiwan

Yao Lin Ong

Department of Accounting and Information System

Chang Jung Christian University

Kway-Jen, Tainan 711

Taiwan

Abstract We decompose the classical Euler sum into a linear combination of sums of the form $H_{p,q}(a;b,c)= \sum_{k=0}^{\infty} \frac{1}{(ak+c)^{q}} \sum_{j=0}^{\tilde{k}} \frac{1}{(aj+b)^{p}} \quad (\tilde{k} = k \;$ if $\; b \leq c, \tilde{k} = k-1 \;$ if $\; b > c)$ which we call Euler sums on arithmetic progressions. Through basic linear relations among these new Euler sums, we are able to evaluate the family $D_{p,q}^{(a)}:=\sum_{b=1}^a H_{p,q}(a;b,b)$ when the weight p + q is odd. In addition, we obtain the evaluation of $T_{1,n}^{(a)}$ no matter when the weight is even or odd and construct a lot of new families which can be evaluated when the weight is odd.

Keywords Euler sums, Kronecker limit formula, Hurwitz zeta function

Classification (MSC2000) Primary: 11M06; Secondary: 11M35, 33B15