Asymptotic estimates for Stieltjes constants: a probabilistic approach

Abstract

Let (g n ) n≥0 be the sequence of Stieltjes constants appearing in the Laurent expansion of the Riemann zeta function. We obtain explicit upper bounds for |g n |, whose order of magnitude is exp n log log n − n 2 log 2 n 1 + O 1 log n as n tends to infinity. To do this, we use a probabilistic approach based on a differential calculus for the gamma process.

Estimates of generalized Stieltjes constants with a quasi-geometric rate of decay

“…e m m n+1 n + 1 m + 1 + 1 n + 1 log n+1 (m + 1) , (1.3) where m = n(1 − 1/ log n) and x stands for the integer part of x. Such a bound follows by gathering together the results by Matsuoka (1985) and Adell (2011). The order of magnitude of the right-hand side in (1.3) is exp n log log n − n 2 log 2 n 1 + O 1 log n , n → ∞.…”

Acceleration Methods for Series: A Probabilistic Perspective

“…As we will see in the following sections, this probabilistic methodology gives a unified approach to deal with various problems in analytic number theory. In this regard, differentiation formulas for linear operators represented by gamma processes have been used in [2] to obtain asymptotic estimates for Stieltjes constants, as well as to compute each Stieltjes constant in the classical spirit of Stieltjes and Berndt [9] (c.f. [3]).…”

Corrigendum to “Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only” [J. Number Theory 158 (2016) 365–396]

“…Writings Γ(z) and ζ(z) denote respectively the gamma and the zeta functions of argument z. The Pochhammer symbol (z) n , which is also known as the generalized factorial function, is defined as the rising factorial 7,8 For sufficiently large n, not necessarily integer, the latter can be given by this useful approximation 5 In particular γ 1 = −0.07281584548 . .…”

“…3-6]. 7 For nonpositive and complex n, only the latter definition (z) n ≡ Γ(z + n)/Γ(z) holds. 8 Note that some writers (mostly German-speaking) call such a function faculté analytique or Facultät, see e.g.…”