An effective asymptotic formula for the Stieltjes constants

Knessl, Charles; Coffey, Mark

doi:10.1090/S0025-5718-2010-02390-7

An effective asymptotic formula for the Stieltjes constants
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by Charles Knessl and Mark W. Coffey PDF

Math. Comp. 80 (2011), 379-386 Request permission

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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