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- W1601388632 abstract "The zeta and related functions are explicitly known at either even values as for the ζ(s), λ(s) and η(s) functions or at odd function values as for the Dirichlet β(s) function. In this paper, the values at integers are obtained for the zeta and related functions by applying new analytical tool that naturally arises within a new theoretical setting. This allows a simple and elementary derivation of the function values at integer points from quite a different point of view. It is based on the hypothesis that negative numbers might be beyond infinity, advanced by Wallis and Euler. I will first define the basic concepts of our theory: (1) a new ordering relation on the set of integer numbers; (2) a new class of real functions, called regular, and the definition of sum that extends the classical definition to the case when the upper limit of summation is less than the lower; (3) a set of conditions imposed on regular functions that defines a new regular method for summation of infinite series." @default.
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- W1601388632 date "2010-01-01" @default.
- W1601388632 modified "2023-09-24" @default.
- W1601388632 title "Elementary Evaluation of the Zeta and Related Functions: An Approach From a New Perspective" @default.
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- W1601388632 doi "https://doi.org/10.1063/1.3497834" @default.
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