On the Values of a Certain Dirichlet Series at Rational Integers

The Values of an Euler Sum at the Negative Integers and a Relation to a Certain Convolution of Bernoulli Numbers

“…We consider the function , (1.1) where , (1.2) are the Harmonic numbers and . This function was studied by Apostol-Vu [1] and Matsuoka [5] who provided an analytic extension for all complex values of s and discussed its values and poles at the negative integers. In this note we shall find a relation between the values and the numbers , defined as the convolution.…”

Laurent expansion of harmonic zeta functions

“…Apostol-Vu [2] and Matsuoka [12] provided the meromorphic continuation of this function in C and investigated its values and poles at the negative integers. In particular, the special values at negative even integers are given by Matsuoka's formula 2…”

“…where H = (H n ) is the sequence of harmonic numbers. Apostol-Vu [2] and Matsuoka [12] have shown that this function, called the harmonic zeta function, can be continued as a meromorphic function with a double pole at s = 1, and an infinite number of simple poles at s = 0 and s = 1 − 2k for each integer k ≥ 1. The Ramanujan summation method allows to sum the series n 1 Hn n s for all values of s, which make possible, for each pole s = a, to give an expression of the constant C a in terms of the sum R n 1 Hn n a of the series in the sense of Ramanujan's summation method.…”

Multiple Hurwitz zeta functions

“…He derived the analytic continuation of ζ(s 1 , s 2 ) using the Poisson summation formula. Perhaps not aware of this work, Matsuoka [29] in 1982 derived the analytic continuation of ζ(s, 1) and Apostol and Vu [5] in 1984 again studied the case r = 2. In both papers, the main tool was the Euler -Maclaurin summation formula.…”