An effective asymptotic formula for the Stieltjes constants

Authors:
Charles Knessl and Mark W. Coffey

Journal:
Math. Comp. **80** (2011), 379-386

MSC (2010):
Primary 41A60, 30E15, 11M06

DOI:
https://doi.org/10.1090/S0025-5718-2010-02390-7

Published electronically:
June 9, 2010

MathSciNet review:
2728984

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Stieltjes constants appear in the coefficients in the regular part of the Laurent expansion of the Riemann zeta function about its only pole at . We present an asymptotic expression for for . This form encapsulates both the leading rate of growth and the oscillations with . Furthermore, our result is effective for computation, consistently in close agreement (for both magnitude and sign) for even moderate values of . Comparison to some earlier work is made.

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Additional Information

**Charles Knessl**

Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607-7045

**Mark W. Coffey**

Affiliation:
Department of Physics, Colorado School of Mines, Golden, Colorado 80401

DOI:
https://doi.org/10.1090/S0025-5718-2010-02390-7

Keywords:
Stieltjes constants,
Riemann zeta function,
Laurent expansion

Received by editor(s):
September 25, 2009

Received by editor(s) in revised form:
November 2, 2009

Published electronically:
June 9, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.