StieltjesGamma

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[γ, n]) is a nonanalytical function which is defined only for nonnegative integer n.

Symmetries and periodicities

Symmetry

No symmetry

Periodicity

No periodicity

Series representations

Other series representations

K. Maslanka: An Explicit Formula Relating Stieltjes Numbers and Li's Numbers  math.NT/0406312 ({}[[1,1]])
{}[[1,1]]

Integral representations

Contour integral representations

Limit representations

Generating functions

Identities

Relation to Li's numbers

K. Maslanka: An Explicit Formula Relating Stieltjes Numbers and Li's Numbers  math.NT/0406312 ({}[[1,1]])
{}[[1,1]]

Inequalities

Theorems

Riemann hypothesis

The Riemann hypothesis is equalivalent to (Li 1997) (Subscript[λ, n])>0 for all integer n>0. The

(Subscript[λ, n])=(1/((n-1)!))((∂^n)/(∂(s^n)))((s^(n-1))log(ζ(s)))(Subscript[|, s==1])

contain derivatives of the Riemann zeta function at s==1 and can be expressed through Stieltjes constants.

History

– E. Cahen (1894)

Copyright