与Zeta函数有关的级数

Series Relating to Riemann Zeta Functions

首先通过考察Zeta函数ζ(p)的余项rn(p)的等价无穷小量,用Abel求和公式给出级数Σ∞n=1nmrn(p)的和.其次借助psi函数和Trigamma函数给出级数Σ∞n=1nan(n-ζ(2)-ζ(3)-..-ζ(n))(|a|<2)的和,并着重考虑了a=1的极限.

Firstly by examining the equivalent infinitesimal quantity of the remainder r n(p) of Zeta function ζ(p) , this paper is concerned with the summation of the series of ∑ ∞ n=1 n mr n(p) by means of Abel summation formula. Secondly, the summation of ∑ ∞ n=1 na n(n-ζ(2)-ζ(3)-…-ζ(n))(|a|<2) is given by using of psi and Trigamma functions. Especially, the limit at a=1 is obtained.