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An asymptotic form for the Stieltjes constants $ \gamma_k(a)$ and for a sum $ S_\gamma(n)$ appearing under the Li criterion


Authors: Charles Knessl and Mark W. Coffey
Journal: Math. Comp. 80 (2011), 2197-2217
MSC (2010): Primary 41A60, 30E15, 11M06
DOI: https://doi.org/10.1090/S0025-5718-2011-02497-X
Published electronically: May 11, 2011
MathSciNet review: 2813355
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Abstract: We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants $ \gamma_k(a)$. These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials $ L_n^\alpha$ to investigate a certain sum $ S_\gamma(n)$ involving the constants $ \gamma_k(1)$ that appears in application of the Li criterion for the Riemann hypothesis. We confirm the sublinear growth of $ S_\gamma(n)+n$, which is consistent with the validity of the Riemann hypothesis.


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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
Email: knessl@uic.edu

Mark W. Coffey
Affiliation: Department of Physics, Colorado School of Mines, Golden, Colorado 80401
Email: mcoffey@mines.edu

DOI: https://doi.org/10.1090/S0025-5718-2011-02497-X
Keywords: Li criterion, Riemann hypothesis, Stieltjes constants, Hurwitz zeta function, Riemann zeta function, Laurent expansion, asymptotic form
Received by editor(s): June 18, 2010
Received by editor(s) in revised form: September 28, 2010
Published electronically: May 11, 2011
Article copyright: © Copyright 2011 American Mathematical Society