An asymptotic form for the Stieltjes constants 𝛾_{𝑘}(𝑎) and for a sum 𝑆ᵧ(𝑛) appearing under the Li criterion

Knessl, Charles; Coffey, Mark

doi:10.1090/S0025-5718-2011-02497-X

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.

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