An asymptotic form for the Stieltjes constants 𝛾_{𝑘}(𝑎) and for a sum 𝑆ᵧ(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 γ k ( a ) \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 α L_n^\alpha to investigate a certain sum S γ ( n ) S_\gamma (n) involving the constants γ k ( 1 ) \gamma _k(1) that appears in application of the Li criterion for the Riemann hypothesis. We confirm the sublinear growth of S γ ( n ) + n S_\gamma (n)+n , which is consistent with the validity of the Riemann hypothesis.