An Asymptotic Form for the Stieltjes Constants (Gamma K)(A) and for A Sum S(Gamma)(N) Appearing Under the Li Criterion

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 γk(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 Lαn to investigate a certain sum Sγ(n) involving the constants γk(1) that appears in application of the Li criterion for the Riemann hypothesis. We confirm the sublinear growth of Sγ(n)+n, which is consistent with the validity of the Riemann hypothesis.