Alternating Euler sums at the negative integers.

Khristo Boyadzhiev (1) , H. Gopalkrishna Gadiyar (2) , R Padma (2)

1 Mathematics Department, The Ohio State University
2 KBC research Center

We study three special Dirichlet series, two of them alternating, related to the Riemann zeta-function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers.

Mots clés: Euler-Boole formula; Euler sum; Hankel contour integration; Bernoulli number; Euler Polynomial; Euler-Maclaurin summation formula; Dirichlet series; Riemann zeta function

Domaines: Mathématiques [math]

Khristo Boyadzhiev, H. Gopalkrishna Gadiyar, R Padma. Alternating Euler sums at the negative integers.. Hardy-Ramanujan Journal, 2009, Volume 32 - 2009, pp.24-37. ⟨10.46298/hrj.2009.165⟩. ⟨hal-01112352⟩