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On the Evaluation of Euler Sums
Experimental Mathematics, 1994 Euler studied double sums of the form $$ \z(r,s)=\sum_{1\le ...
Crandall, Richard E., Buhler, Joe P.
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Euler Graph Transformations for Euler Diagram Layout [PDF]
, 2010 Euler diagrams are frequently used for visualizing information about collections of objects and form an important component of various visual languages.
Howse, John +7 more
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Properties of Euler Diagrams [PDF]
, 2007 Euler diagrams have numerous application areas, with a large variety of languages based on them. In relation to software engineering, such areas encompass modelling and specification including from a formal perspective. In all of these application areas,
Howse, John +3 more
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A Survey of Euler Diagrams [PDF]
, 2014 Euler diagrams visually represent containment, inpresstersection and exclusion using closed curves. They first appeared several hundred years ago, however, there has been a resurgence in Euler diagram research in the twenty-first century.
Rodgers, Peter
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Restricted sum formula of alternating Euler sums [PDF]
The Ramanujan Journal, 2014 In this paper we study restricted sum formulas involving alternating Euler sums which are defined by ζ(s_1,...,s_{d};ε_1,...,ε_d)=\sum_{n_1>...>n_d\ge 1}\frac{ε_1^{n_1}... ε_{d}^{n_d}}{n_1^{s_1}... n_d^{s_d}}, for all positive integers s_1,...,s_{d} and ε_1=\pm 1,..., ε_{d}=\pm 1 with (s_1,ε_1) unequal (1,1).
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Alternating Euler $T$-sums and Euler $\tilde S$-sums
, 2020 In this paper, we study the alternating Euler $T$-sums and related sums by using the method of contour integration. We establish the explicit formulas for all linear and quadratic Euler $T$-sums and related sums. Some interesting new consequences and illustrative examples are considered.
Wang, Weiping, Xu, Ce
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Parametric Euler sum identities
Journal of Mathematical Analysis and Applications, 2006 12 ...
Borwein, David +2 more
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A Counterexample to Euler's Sum of Powers Conjecture [PDF]
Mathematics of Computation, 1967 Es handelt sich urn eine Vermutung Eulers, die Dickson im zweiten Band seiner Geschichte der Zahlentheorie auf Seite 682 sehr vorsichtig folgendermaßen ausdrückt: ``Euler expressed his belief that no sum of four fifth powers is a fifth power.'' Die Verff.
Lander, L. J., Parkin, T. R.
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Euler Sums and Contour Integral Representations [PDF]
Experimental Mathematics, 1998 This paper develops an approach to the evaluation of Euler sums involving harmonic numbers either linearly or nonlinearly. We give explicit formul{æ} for certain classes of Euler sums in terms of values of the Riemann zeta function at positive integers. The approach is based on simple contour integral representations and residue computations.
Flajolet, Philippe, Salvy, Bruno
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Embedding wellformed Euler diagrams [PDF]
, 2008 Euler diagrams are collections of labelled closed curves. They are often used to represent information about the relationship between sets and, as such, they have numerous applications including: visualizing biological data, diagrammatic logics, and ...
Andrew Fish +8 more
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