A note on some constants related to the zeta-function and their relationship with the Gregory coefficients

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Abstract

In this article, new series for the first and second Stieltjes constants (also known as generalized Euler’s constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so-called Gregory coefficients, which are also known as the (reciprocal) logarithmic numbers, the Cauchy numbers of the first kind and the Bernoulli numbers of the second kind. In addition, two interesting series with rational terms for Euler’s constant \(\gamma \) and the constant \(\ln 2\pi \) are given, and yet another generalization of Euler’s constant is proposed and various formulas for the calculation of these constants are obtained. Finally, we mention in the paper that almost all the constants considered in this work admit simple representations via the Ramanujan summation.

Keywords

Stieltjes constants Generalized Euler’s constants Series expansions Gregory’s coefficients Rational coefficients Harmonic product of sequences 

Mathematics Subject Classification

11M06 41A20 40G99 40C99 41A58 65D20 65D30 

Notes

Acknowledgements

The authors are grateful to Vladimir V. Reshetnikov for his kind help and useful remarks. The authors also thank the referee for his valuable suggestions and comments.

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Authors and Affiliations

  1. 1.University of ToulonToulonFrance
  2. 2.Steklov Institute of Mathematics at St. PetersburgSt. PetersburgRussia
  3. 3.St. Petersburg State University of Architecture and Civil EngineeringSt. PetersburgRussia
  4. 4.Université Côte d’Azur, CNRS, LJAD (UMR 7351)NiceFrance

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