Approximating and bounding fractional Stieltjes constants

A Bound for Stieltjes Constants

“…Hence e α(log w(α)−1/w(α)) = e α (log w(α)−1/w(α)) < (log α) α Thus the main term of Matsuoka's bound follows from (16). , Matsuoka [16] and Saad Eddin [19], our bound and the conjecture from [10] as well as our new bound from Theorem 8.…”

“…Note that the main term of the bound in Theorem 8 differs only by a factor of α log α from the conjectured bound given in [10]: (16) |C α (a)| ≤ 2 e α(log w(α)−1/w(α)) .…”

“…These constants have several interesting and unexpected applications in the zeta function theory, as was shown recently in [4], [3], [8], and [9]. Moreover, the classical Euler-Maclaurin Summation can be used to prove (see [10]) that, if we set C α (a) = γ α (a) − log α (a)…”

“…Figure 3. On a logarithmic scale we show the absolute values of the Stieltjes constants γ α = C α (1), C α (2/3), C α (1/3) and C α (1/10) along with the bounds by Berndt [1], Williams and Zhang[21], Matsuoka[16] and Saad Eddin[19], our bound and the conjecture from[10] as well as our new bound from Theorem 8.…”

“…the bound by Matsuoka[16] which holds for m > 4: |γ m | < 10 −4 (log m) m (4) the bound by Saad Eddin[19]: Let θ(m) = m+1 log 2(m+1) π − 1 then |γ m | ≤ m! • 2 √ 2e −(n+1) log θ(m)+θ(m)(log θ(m)+log 2 πe ) 1 + 2 −θ(m)−1 θ(m) + 1 θ(m) − 1. our bound from[10]: For α > 0 let x = π 2 e n+1 )Γ(α + 1) (2π) n+1 (n + 1) α+1…”