New results on the Stieltjes constants: Asymptotic and exact evaluation

@article{Coffey2005NewRO,
  title={New results on the Stieltjes constants: Asymptotic and exact evaluation},
  author={Mark W. Coffey},
  journal={Journal of Mathematical Analysis and Applications},
  year={2005},
  volume={317},
  pages={603-612}
}
  • M. Coffey
  • Published 2005
  • Mathematics, Physics
  • Journal of Mathematical Analysis and Applications
Abstract The Stieltjes constants γ k ( a ) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s = 1 . We present new asymptotic, summatory, and other exact expressions for these and related constants. 
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  • Mathematics
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References

SHOWING 1-10 OF 35 REFERENCES
Zeta Values and Differential Operators on the Circle
Abstract The Fock representation of the Virasoro Lie algebra is extended to a larger graded Lie subalgebra of the algebra of differential operators on the circle. The central cocycle is related toExpand
Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series
We prove that a series derived using Euler's transformation provides the analytic continuation of ζ(s) for all complex s ¬= 1. At negative integers the series becomes a finite sum whose value isExpand
Generalized Euler Constants for Arithmetical Progressions
The work of Lehmer and Briggs on Euler constants in arithmetical progressions is extended to the generalized Euler constants that arise in the Laurent expansion of g(s) about s = 1 . The results areExpand
Power series expansions of Riemann’s function
We show how high-precision values of the coefficients of power series expansions of functions related to Riemann's í function may be calculated. We also show how the Stieltjes constants can beExpand
Series Associated with the Zeta and Related Functions
Preface. Acknowledgements. 1. Introduction and Preliminaries. 2. The Zeta and Related Functions. 3. Series Involving Zeta Functions. 4. Evaluations and Series Representations. 5. Determinants of theExpand
The Positivity of a Sequence of Numbers and the Riemann Hypothesis
Abstract In this note, we prove that the Riemann hypothesis for the Dedekind zeta function is equivalent to the nonnegativity of a sequence of real numbers.
Relations and positivity results for the derivatives of the Riemann ξ function
We present and evaluate the integer-order derivatives of the Riemann xi function. These derivatives contain logarithmic integrals of powers multiplying a specific Jacobi theta function and as suchExpand
Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions
This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability lawsExpand
The Theory of the Riemann Zeta-Function
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspectsExpand
Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants
TLDR
The numerical method uses Newton-Cotes integration formulae for very high-degree interpolating polynomials; it differs in implementation from, but compares in error bounding to, Euler-Maclaurin summation based methods. Expand
...
1
2
3
4
...