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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
Published electronically: June 9, 2010
MathSciNet review: 2728984
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Abstract: The Stieltjes constants $ \gamma_k$ appear in the coefficients in the regular part of the Laurent expansion of the Riemann zeta function $ \zeta(s)$ about its only pole at $ s=1$. We present an asymptotic expression for $ \gamma_k$ for $ k \gg 1$. This form encapsulates both the leading rate of growth and the oscillations with $ k$. Furthermore, our result is effective for computation, consistently in close agreement (for both magnitude and sign) for even moderate values of $ k$. 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

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.