# Euler sums on arithmetic progressions

### From NZJM

**New Zealand Journal of Mathematics**

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

**Minking Eie**

Department of Mathematics

National Chung Cheng University

Minhsiung, Chiayi 62145

Taiwan

mailto:minking@math.ccu.edu.tw

**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 if if
which we call Euler sums on arithmetic progressions. Through basic linear relations among
these new Euler sums, we are able to evaluate the family
when the weight *p* + *q* is odd. In addition, we obtain the evaluation of 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