منابع مشابه
Consecutive Evaluation of Euler Sums
is the Riemann zeta function. The numbers S(r ,p) for p+r odd were explicitly evaluated for the first time in [3]. As commented in [3], there is strong evidence that forp+r even S(r ,p) can be evaluated only in some particular cases: S(p,p), S(2,4), S(4,2). A powerful method for evaluation of Euler sums, based on the residue theorem, was presented in [7]. The purpose of this note is to describe...
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In response to a letter from Goldbach, Euler considered sums of the form ∞ ∑ k=1 ( 1 + 1 2m + · · ·+ 1 km ) (k + 1)−n for positive integers m and n. Euler was able to give explicit values for certain of these sums in terms of the Riemann zeta function. In a recent companion paper, Euler’s results were extended to a significantly larger class of sums of this type, including sums with alternating...
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In response to a letter from Goldbach, Euler considered sums of the form where s and t are positive integers. As Euler discovered by a process of extrapolation (from s + t 13), h (s; t) can be evaluated in terms of Riemann-functions when s + t is odd. We provide a rigorous proof of Euler's discovery and then give analogous evaluations with proofs for corresponding alternating sums. Relatedly we...
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ژورنال
عنوان ژورنال: International Journal of Mathematics and Mathematical Sciences
سال: 2002
ISSN: 0161-1712,1687-0425
DOI: 10.1155/s0161171202007871