منابع مشابه
The Evaluation of Tornheim Double Sums
We provide an explicit formula for the Tornheim double series in terms of integrals involving the Hurwitz zeta function. We also study the limit when the parameters of the Tornheim sum become natural numbers, and show that in that case it can be expressed in terms of definite integrals of triple products of Bernoulli polynomials and the Bernoulli function Ak(q) := kζ(1− k, q).
متن کاملEuler and the Partial Sums of the Prime Harmonic Series
Abstract. In a 1737 paper, Euler gave the first proof that the sum of the reciprocals of the prime numbers diverges. That paper can be considered as the founding document of analytic number theory, and its key innovation — socalled Euler products — are now ubiquitous in the field. In this note, we probe Euler’s claim there that “the sum of the reciprocals of the prime numbers” is “as the logari...
متن کاملOn Mordell-tornheim Sums and Multiple Zeta Values
RÉSUMÉ. Nous prouvons que toute somme de Mordell-Tornheim avec des arguments entiers positifs peut s’écrire comme une combinaison linéaire rationnelle de valeurs prises par des fonctions multi-zêta ayant le même poids et la même profondeur. Selon un résultat de Tsumura, il s’ensuit que toute somme de Mordell-Tornheim ayant un poids et une profondeur de parité différente peut s’exprimer comme un...
متن کاملComputation and theory of extended Mordell-Tornheim-Witten sums
We consider some fundamental generalized Mordell–Tornheim–Witten (MTW) zeta-function values along with their derivatives, and explore connections with multiplezeta values (MZVs). To achieve this, we make use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function theory. Our original motivation was to represent unresolved constructs...
متن کاملComputation and theory of Mordell-Tornheim-Witten sums II
In [7] the current authors, along with the late and much-missed Richard Crandall (1947– 2012), considered generalized Mordell–Tornheim–Witten (MTW) zeta-function values along with their derivatives, and explored connections with multiple-zeta values (MZVs). This entailed use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function th...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2017
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2016.10.002