On the Derivative of a Zeta Function at Zero

• Main Authors: Kuo-Chuan Hsieh, 謝國釧
• Other Authors: Minking Eie
• Format: Others
• Language: en_US
• Published: 2001
• Online Access: http://ndltd.ncl.edu.tw/handle/40527658617740684564

碩士 === 國立中正大學 === 數學研究所 === 89 === We present a new, simple difference equation (as at s＝0) can be used to control the analytic continuation of a Dirichlet series without using anything more than the original convergent series. We begin with the generalized zeta function (or the Hurwitz zeta function).Our concern is to establish the following beautiful formula relating the Gamma function and the Hurwitz zeta function. What is more, let P(x) be a product of linear forms with real coefficients; consider the zeta function Z(P;s) associated with P(x). In 1990, as given in [1], there was a result about Z(P;s)associated with P(x). Z(P;s) was represented by zeta function and Gamma functions. Furthermore he has found explicit formulas for Z(P;－m), m＝0, 1, 2, 3,…, and the derivative of Z(P;s) at s=0; the latter under the condition that all zero points of the polynomial P(x) are in the unit circle. Finally, we show that the meromorphic continuation and Eie's formula for Z(P;0)and the derivative of Z(P;s) at s=0 still hold under the obvious assumption that all zero points of P(x) are not positive integers. Suitably rewriting the zeta function Z(P;s), we reduce, after some computations, the proof of my general case to Eie's.