# 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} }

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.

#### 44 Citations

A New Effective Asymptotic Formula for the Stieltjes Constants

- Mathematics
- 2014

We derive a new integral formula for the Stieltjes constants. The new formula permits easy computations as well as an exact approximate asymptotic formula. Both the sign oscillations and the leading… Expand

Double series expression for the Stieltjes constants

- Mathematics, Physics
- 2010

Abstract We present expressions in terms of a double infinite series for the Stieltjes constants γk(a). These constants appear in the regular part of the Laurent expansion for the Hurwitz zeta… Expand

Asymptotic estimates for Stieltjes constants: a probabilistic approach

- Mathematics
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2010

Let (γn)n ≥ 0 be the sequence of Stieltjes constants appearing in the Laurent expansion of the Riemann zeta function. We obtain explicit upper bounds for |γn|, whose order of magnitude is as n tends… Expand

The difference between two Stieltjes constants

- Mathematics
- 2009

The Stieltjes constants are the coefficients of the Laurent expansion of the Hurwitz zeta function and surprisingly little is known about them. In this paper we derive some relations for the… Expand

Addison-type series representation for the Stieltjes constants

- Mathematics, Physics
- 2009

The Stieltjes constants γk(a) appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about its only pole at s=1. We generalize a technique of… Expand

An effective asymptotic formula for the Stieltjes constants

- Computer Science, Mathematics
- Math. Comput.
- 2011

An asymptotic expression for γ k for k >> 1 is presented, which encapsulates both the leading rate of growth and the oscillations with k, and is effective for computation. Expand

An asymptotic form for the Stieltjes constants gammak(a) and for a sum Sgamma(n) appearing under the Li criterion

- Computer Science, Mathematics
- Math. Comput.
- 2011

Asymptotic results for the Laguerre polynomials Lαn are used to investigate a certain sum Sγ(n) involving the constants γk(1) that appears in application of the Li criterion for the Riemann hypothesis. Expand

Some applications of the Stieltjes constants

- Mathematics
- 2009

In this paper we present some applications of the Stieltjes constants including, for example, new derivations of Binet's formulae for the log gamma function and the evaluation of some integrals… Expand

Note on the Stieltjes constants: series with Stirling numbers of the first kind

- Mathematics
- 2016

The Stieltjes constants $\gamma_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about $s=1$. We generalize the integral and… Expand

Series representations for the Stieltjes constants

- Mathematics, Physics
- 2009

The Stieltjes constants \gamma_k(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function \zeta(s,a) about s=1. We present series representations of… Expand

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