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

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- by Charles Knessl and Mark W. Coffey PDF
- Math. Comp.
**80**(2011), 2197-2217 Request permission

## 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.## References

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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
- Received by editor(s): June 18, 2010
- Received by editor(s) in revised form: September 28, 2010
- Published electronically: May 11, 2011
- © Copyright 2011 American Mathematical Society
- Journal: Math. Comp.
**80**(2011), 2197-2217 - MSC (2010): Primary 41A60, 30E15, 11M06
- DOI: https://doi.org/10.1090/S0025-5718-2011-02497-X
- MathSciNet review: 2813355