Introduction to the Bernoulli function
Peter Luschny

A companion to arXiv:2009.06743v2 [math.HO]

Numbering of formulas as in version 2.

Browsable at BernoulliFunctionNotebook.

Source at BernoulliFunctionSource.

Needs Wolfram Language kernel for Jupyter. See instructions at GitHub.


Stieljes constants and zeta function

Formula 1

Laurent series of the Riemann zeta function

Formula 2

Stieltjes constants, defined via integral

Bernoulli constants and Bernoulli function

Formula 3

Bernoulli constants, defined via integral

Memoization of the real part of the Bernoulli constants with single precision. (For quick checking and plotting, for higher precision use the two parameter form.)

Formula 4

Bernoulli function, definition.

Formula 5

Bernoulli function using the Riemann zeta representation.

From now onwards we use Mathematica's efficient implementation of the Zeta function for the Bernoulli function.

Bernoulli function on the critical line and the zeta zeros which are the same as the Bernoulli zeros.

Bernoulli function rises up at Riemann's critical line.