﻿ The Bernoulli Confusion

The Bernoulli Confusion

Based on a long discussion concerning the Bernoulli numbers in the newsgroup de.sci.mathematik in the year 2004:

• An exposé of my contributions and the summary of the most important contributions of group. No longer up-to-date in every aspect, but authentic.

• I put the central question of the discussion to Donald E. Knuth. My open letter and Knuth's reply.

• My response to Knuth, which I never mailed. Here I try to ponder his arguments and set out the alternative. Read this paper if you want to understand the Bernoulli rebellion.

What is it all about? We look at the function

 B(s) = -s ζ(1 - s)

where zeta denotes the Riemann zeta function. Note that B(0) and B(1) are well defined by the limit. The Bernoulli numbers then are defined as the values of this function at the nonnegative integers, thus as B(n). Is this a sensible definition?

The Bernoulli Function

The Bernoulli function B(s) = -s zeta(1 - s).

The Bernoulli Riddle

Look at the following beginning of a sequence of rational numbers:

 B0  = 1 B1  = B2  = 1 - 1/2 - 1/3 B4  = 1 - 1/2 - 1/3 - 1/5 B6  = 1 - 1/2 - 1/3 - 1/7 B8  = 1 - 1/2 - 1/3 - 1/5 B10 = 1 - 1/2 - 1/3 - 1/11

The question is: What value receives B1? This question was answered in 1840 by two grand masters of the Bernoulli numbers, von Staudt and Clausen, independently. The right solution is not difficult to find, try it. And sure, the prime numbers govern the picture. You don't need to be a rocket scientist to see all this. In fact, you must be a rocket scientist to be able not to see it. Or at least a mathematician who only copies from other work, for example from the Handbook of the American National Institute of Standards. Astounding, even unbelievable is what an absurd answer you can find there. Believe it or not they say B1 = −1 + 1/2. No explanation given.

Hear us, O Mathematicians of the World!

B1 = 1 - 1/2

The first chorus, from the left: Unbelievable! We never thought this could be true! The second chorus, from the right: Don't believe this. This will only create confusion! ... the greatest confusion of the century! But we say:

The Bernoulli numbers are the children of the zeta function ...

And G. H. Hardy reminds us: "The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way."

Jakob Bernoulli says:

The Bernoulli Rebellion

But No, No, Ten Thousand Times No!

These are the words of D. E. Knuth in his famous paper Two Notes on Notation by which he objects to the opinion that 00 is to be left undefined. And these also shall be our words when we battle the absurd B1 = -1/2, in the spirit of Jakob's 'invito patre'.

Junk the chapter on the Bernoulli numbers in the Handbook of Mathematical Functions. And write an email to Karl Dilcher, the editor of the Bernoulli chapter in the Digital Library of Mathematical Functions in the (probable) case that he will carry on this unhappy tradition in this online library and object! Edited: Well, in the meantime DLMF is out and the sad truth is that Dilcher continued the unhappy tradition of the HMF.

Nevertheless, it remains clear that the definition based on the zeta function will win in the long run over the ad-hoc definition of the Handbook and the DLMF; all Dilcher achieved was to prolong the pains of confusion.

Bernoulli numbers and special numbers

Read the Wikipedia article on Bernoullli numbers where you can see, among other things, that:

• The important connection with Stirling polynomials gives
B1n(1)=1/2.
• The natural connection with the Stirling set numbers gives
B1 = 1/2.
• The beautiful connection with the Worpitzky numbers gives
B1 = 1/2.
• The connection with the Stirling cycle number gives
B1 = 1/2.
• The Akiyama-Tanigawa algorithm gives
B1 = 1/2.
• The important connection with the Eulerian numbers "are valid for n ≥ 0 if B1 is set to 1/2. If B1 is set to −1/2 they are valid only for n ≥ 1 and n ≥ 2 respectively."

Let An(x) denote Eulerian polynomials, En(x) Euler polynomials,  let Bn(x) denote the Bernoulli polynomials and ζ(n) the Riemann Zeta function. {n over k} denotes the Stirling numbers of the second kind.

Look at this list of wonderful formulas. Anything wrong? No. They are correct. What is wrong is that they do not involve the Bernoulli numbers. Allegedly! Just for one single reason: because B1 is supposed to be −1/2. What a shame! The truth is of course that they are part of this amazing chain in the hart of concrete mathematics. Because the Bernoulli numbers are Bn(1) (and not Bn(0), from which they differ (because of symmetry) just in the case n = 1).

But how irritating and misleading this is. Just imagine the thousands and thousands of workarounds in mathematical formulas which are forced upon us by using the wrong definition in the connection with the Eulerian polynomials, the Euler polynomials, the Stirling numbers, the Worpitzky numbers, the zeta function and uncounted binomial sums.

Bernoulli numbers and expansions

No, the mathematical community should not allow the authorities and Institutes of Standards to get away with this balderdash to distort the sign of B1. And π (Pi) does not equal 3.2, even if enforced by Pi-bill ...

Bernoulli numbers and polynomials

Bernoulli polynomials are not at all the only polynomials Pn(x) with Pn(1) = Bn. Two other polynomial sequences are

• the Stirling polynomials and
• the Zeta polynomials.

For the definition of the Stirling polynomials σn(x) see formula (6.52) in Concrete Mathematics by Graham, Knuth and Patashnik. The Zeta polynomials were introduced in Zeta Polynomials and Harmonic Numbers,

 (I)

Here ζ(s) is the Riemann zeta function.

Let Hn denote the harmonic number. Since ζn+1(1) = −(n+1)ζ(−n) we have for n > 0

 (II)

I could not find this representation of the Bernoulli numbers in the literature. So I showed the formula in the newsgroup de.sci.mathematik; however, no one could provide a reference. If you know a reference for this formula, please let me know — otherwise I might erroneously assume that I was the first to have found this beautiful connection between the Bernoulli numbers and the harmonic numbers. For a more general setup see: A sequence transformation and the Bernoulli numbers.

Jakob, is there any hope?

Donald Knuth writes in his respons:

"Today’s long-standing mathematical conventions have many, many defects, and I can think of dozens of cases where changes would improve the current situation and make it easier on all future mathematicians."

Let us not wait longer!

The Euler-Riemann reflection formula of the zeta function belongs to the crown jewels of mathematics. Let us no longer allow the barbarism to violate this equation by giving B1 the value -1/2. And also let us honour the great Jakob Bernoulli by simply no longer fasten on him numbers which he has never written. Is it not conceivable that he had recognized the connection with the zeta function, at least on some intuitive level, whereas some of his followers were not in the position to see this?