| Issue |
EPJ Web Conf.
Volume 244, 2020
Complexity and Disorder Meetings 2018-2019-2020
|
|
|---|---|---|
| Article Number | 01008 | |
| Number of page(s) | 5 | |
| DOI | https://doi.org/10.1051/epjconf/202024401008 | |
| Published online | 15 October 2020 | |
https://doi.org/10.1051/epjconf/202024401008
Zeta-regularization of arithmetic sequences
CNRS, IMJ-PRG, Sorbonne Université 4 Place Jussieu F-75252 Paris Cedex 05 France
* e-mail: jean-paul.allouche@imj-prg.fr
Published online: 15 October 2020
Is it possible to give a reasonable value to the infinite product 1 × 2 × 3 × · · · × n × · · · ? In other words, can we define some sort of convergence of the finite product 1 × 2 × 3 × · · · × n when n goes to infinity? One way is to relate this product to the R√iemann zeta function and to its analytic continuation. This approach leads to: 1 × 2 × 3 × · · · × n × · · · = 2π. More generally the “zeta-regularization” of an infinite product consists of introducing a related Dirichlet series and its analytic continuation at 0 (if it exists). We will survey some properties of this generalized product and allude to applications. Then we will give two families of possibly new examples: one unifies and generalizes known results for the zeta-regularization of the products of Fibonacci, balanced and Lucas-balanced numbers; the other studies the zeta-regularized products of values of classical arithmetic functions. Finally we ask for a possible zeta-regularity notion of complexity.
© The Authors, published by EDP Sciences, 2020
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