Proceedings of the Japan Academy, Series A, Mathematical Sciences

Applications of the Laurent-Stieltjes constants for Dirichlet $L$-series

Sumaia Saad Eddin

Full-text: Open access

Abstract

The Laurent-Stieltjes constants $\gamma_{n}(\chi)$ are, up to a trivial coefficient, the coefficients of the Laurent expansion of the usual Dirichlet $L$-series: when $\chi$ is non-principal, $(-1)^{n}\gamma_{n}(\chi)$ is simply the value of the $n$-th derivative of $L(s,\chi)$ at $s=1$. In this paper, we give an approximation of the Dirichlet $L$-functions in the neighborhood of $s=1$ by a short Taylor polynomial. We also prove that the Riemann zeta function $\zeta(s)$ has no zeros in the region $|s-1|\leq 2.2093$, with $0\leq \Re{(s)}\leq 1$. This work is a continuation of [24].

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 93, Number 10 (2017), 120-123.

Dates
First available in Project Euclid: 30 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.pja/1512032605

Digital Object Identifier
doi:10.3792/pjaa.93.120

Mathematical Reviews number (MathSciNet)
MR3732901

Zentralblatt MATH identifier
06850986

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11Y60: Evaluation of constants

Keywords
The Laurent-Stieltjes constants Dirichlet $L$-function Riemann zeta function

Citation

Saad Eddin, Sumaia. Applications of the Laurent-Stieltjes constants for Dirichlet $L$-series. Proc. Japan Acad. Ser. A Math. Sci. 93 (2017), no. 10, 120--123. doi:10.3792/pjaa.93.120. https://projecteuclid.org/euclid.pja/1512032605


Export citation

References

  • J. A. Adell, Asymptotic estimates for Stieltjes constants: a probabilistic approach, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2128, 954–963.
  • J. A. Adell and A. Lekuona, Fast computation of the Stieltjes constants, Math. Comp. 86 (2017), no. 307, 2479–2492.
  • A. Berger, Sur une sommation de quelques séries, Nova. Acta Reg. Soc. Ups. 12 (1883), 31 pp.
  • B. C. Berndt, On the Hurwitz zeta-function, Rocky Mountain J. Math. 2 (1972), no. 1, 151–157.
  • W. E. Briggs, Some constants associated with the Riemann zeta-function, Michigan Math. J. 3 (1955–1956), 117–121.
  • W. E. Briggs and S. Chowla, The power series coefficients of $\zeta(s)$, Amer. Math. Monthly 62 (1955), 323–325.
  • M. W. Coffey, Hypergeometric summation representations of the Stieltjes constants, Analysis (Munich) 33 (2013), no. 2, 121–142.
  • M. W. Coffey, Series representations for the Stieltjes constants, Rocky Mountain J. Math. 44 (2014), no. 2, 443–477.
  • C. Deninger, On the analogue of the formula of Chowla and Selberg for real quadratic fields, J. Reine Angew. Math. 351 (1984), 171–191.
  • R. P. Ferguson, An application of Stieltjes integration to the power series coefficients of the Riemann zeta function, Amer. Math. Monthly 70 (1963), 60–61.
  • M. Gut, Die Zetafunktion, die Klassenzahl und die Kronecker'sche Grenzformel eines beliebigen Kreiskörpers, Comment. Math. Helv. 1 (1929), no. 1, 160–226.
  • H. Ishikawa, On the coefficients of the Taylor expansion of the Dirichlet $L$-function at $s=1$, Acta Arith. 97 (2001), no. 1, 41–52.
  • M. I. Israilov, The Laurent expansion of the Riemann zeta function, Trudy Mat. Inst. Steklov. 158 (1981), 98–104, 229.
  • J. L. W. V. Jensen, Sur la fonction $\zeta (s)$ de Riemann, C. R. Acad. Sci. Paris 104 (1887), 1156–1159.
  • S. Kanemitsu, On evaluation of certain limits in closed form, in Théorie des nombres (Quebec, PQ, 1987), 459–474, de Gruyter, Berlin, 1989.
  • J. C. Kluyver, On certain series of Mr. Hardy, Quart. J. Pure Appl. Math. 50 (1927), 185–192.
  • C. Knessl and M. W. Coffey, An effective asymptotic formula for the Stieltjes constants, Math. Comp. 80 (2011), no. 273, 379–386.
  • R. Kreminski, Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants, Math. Comp. 72 (2003), no. 243, 1379–1397.
  • E. Lammel, Ein Beweis, dass die Riemannsche Zetafunktion $\zeta (s)$ in $|s-1|\leqq 1$ keine Nullstelle besitzt, Univ. Nac. Tucumán Rev. Ser. A 16 (1966), 209–217.
  • M. Lerch, Sur quelques formules relatives au nombre des classes, Bull. Sci. Math. 21 (1897), 290–304.
  • Y. Matsuoka, Generalized Euler constants associated with the Riemann zeta function, in Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984), 279–295, World Sci. Publishing, Singapore, 1985.
  • S. Ramanujan, Collected papers of Srinivasa Ramanujan, AMS Chelsea Publishing, Providence, RI, 2000.
  • S. Saad Eddin, Two problems with Laurent-Stieltjes coefficients, LAP Lambert Academic Publishing, Saarbrücken, 2017.
  • S. Saad Eddin, Explicit upper bounds for the Stieltjes constants, J. Number Theory 133 (2013), no. 3, 1027–1044.
  • S. Saad Eddin, On two problems concerning the Laurent-Stieltjes coefficients of Dirichlet $L$-series, 2013, Ph. D thesis. (University of Lille 1, France).
  • M. Toyoizumi, On the size of $L^{(k)}(1,\chi)$, J. Indian Math. Soc. (N.S.) 60 (1994), no. 1–4, 145–149.
  • D. P. Verma, Laurent's expansion of Riemann's zeta-function, Indian J. Math. 5 (1963), 13–16.
  • N. Y. Zhang and K. S. Williams, Some results on the generalized Stieltjes constants, Analysis 14 (1994), no. 2–3, 147–162.