User:Peter Luschny/EulerConstantsAndBernoulliFunction

Generalized Euler constants and the Bernoulli function

K E Y W O R D S: Bernoulli numbers, Euler constant, Stieltjes constants, Franel-Blagouchine constants, Riemann zeta function, Bernoulli function, Laurent series expansion.

Concerned with sequences:
Sti(0) = A001620 (Euler's constant gamma) (cf. A262235/A075266),
Sti(1) = A082633 (gamma_1) (cf. A262382/A262383),
Sti(2) = A086279 (gamma_2) (cf. A262384/A262385),
Sti(3) = A086280 (gamma_3) (cf. A262386/A262387),
Sti(4) = A086281 (gamma_4), Sti(5) = A086282 (gamma_5),
Sti(6) = A183141 (gamma_6), Sti(7) = A183167 (gamma_7),
Sti(8) = A183206 (gamma_8), Sti(9) = A184853 (gamma_9),
Sti(10) = A184854 (gamma_10).
Cf. A061444, A059956, A073002.

New sequences: 
Sti(1/2) = A301816, Sti(3/2) = A301817.
Cf. A301813, A301814, A301815, A303638, A306016.

Introduction

Our blog post investigates the relationship between the Bernoulli function and the generalized Euler constants.

Euler discovered the constant $\scriptstyle \gamma$ (A001620) in a study of the harmonic numbers. Euler tried hard to reduce this constant to other known values and reconsidered it often over his lifetime but did not succeed; in his own words:

"This number seems also the more noteworthy because even though I have spent much effort in investigating it, I have not been able to reduce it to a known value."

In these studies he developed approximation techniques and other numerical methods which are still in use today. In 1765, while studying the Gamma function, he derives the formula

$\Gamma '(1)=-\gamma .$

In fact in the same paper (E393) he also gives a more complicated formula relating $\scriptstyle \gamma$ to the Bernoulli numbers (here in the form as reconstructed by J. C. Lagarias):

$\gamma =\int _{0}^{\infty }\left(\sum _{n=1}^{\infty }\left(\operatorname {B} _{n}-(-1)^{n}\right){\frac {t^{n-1}}{n!}}\right)dt$

This was certainly the first time that a connection has been made between $\scriptstyle \gamma$ and the Bernoulli numbers. In this blog post we consider a kind of generalization of these two formulas:

$\operatorname {B} '(0)=-\gamma .$

Here $\scriptstyle \operatorname {B} (x)$ is the Bernoulli function introduced in the Bernoulli Manifesto, which reflects a discussion the author had with Don Knuth some years ago.

This formula shows that $\scriptstyle \gamma$ can be seen as the speed of the real Bernoulli function when entering the positive half plane after descending from the negative infinity as a negative-super-exponential function (i.e. $\scriptstyle f(0)=1$ and for all $\scriptstyle x,y\leq 0:\ f(x)f(y)\geq f(x+y)$ ), before the up to then monotonous function turns into a strongly fluctuating one, marked by the Bernoulli numbers at the integers.

I spared the effort to convert the post from TeX to Html, but if you like to know more about the connection between the Euler's constant, its generalization by Stieltjes and the Bernoulli function you can read it here:

Generalized Euler constants and the Bernoulli function.

 

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P. S. Help needed: I observed that apparently no translation of Riemann's classical paper "Über die Anzahl der Primzahlen unter einer gegebenen Größe" into English has a free license. Therefore I translated it myself and released the translation worldwide into the public domain. However, the translation still needs native English speakers to improve it. At the bottom line of this page you find a link which shows how you can help: Riemann's paper.

(By the way in this paper Riemann also proves that the Bernoulli function has the same complex zeros as the Zeta function (although he did not use this terminology). So if you want to prove Riemann's hypothesis it is sufficient to understand the Bernoulli function.)