Stieltjes constants

The Stieltjes constants are a sequence of real numbers which are defined by the following limits:

Where is the Euler constant. It is believed that are irrational. Evidence of this could not yet be provided. Due to their definition, they are sometimes referred to as generalized Euler constants. They occur in the Laurent expansion of the Riemann zeta function

And in the evaluation of certain definite integrals on:

They are closely related to the numbers

Together. These can be numerically well over a convergence acceleration ( continued averaging) to calculate. It is the recursion

And the explicit representation using the Bernoulli numbers:

From the recursion is obtained for n = 1, the identity, that is, for Euler's constant, the alternating series

The number of Vacca is very similar.

The result shows an oscillating behavior with asymptotically approaches 0 slowly declining " frequency ". It is known to

Applies.

Numeric values

Generalization

For the Hurwitz zeta function is of importance: