Saturday, October 27, 2012

The Bernoulli numbers and the primes


News (21 oct 2012):

Shi Bai and I would like to announce that we have factored the
remaining C173 of the numerator of the 202th Bernoulli number (see
http://homes.cerias.purdue.edu/~ssw/bernoulli/bnum for a list of known
factors).

We used the CADO-NFS program (http://cado-nfs.gforge.inria.fr/). The
polynomial selection took around 10 days on a 48 core 2.2GHz AMD
Opteron. The sieving took around 42 days on the 48 core machine.
Linear algebra took around 15 days on 72 cores on an AMD Opteron
2.3 GHz. The final square root phase consumed less than a day on
8 cores on the same machine.

The C173 factored into a 76-digit prime and a 97-digit prime.


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The Bernoulli numbers Bn play an important role in several topics of mathematics. These numbers can be defined by the power series

where all numbers Bn are zero with odd index n > 1. The even-indexed rational numbers Bn alternate in sign. The first values are

Sum of consecutive integer powers
Jakob Bernoulli introduced a sequence of rational numbers, later called Bernoulli numbers, to compute the sum of consecutive integer powers. This formula is given by

Values of the Riemann zeta function
The Bernoulli numbers are connected with the Riemann zeta function

on the positive real axis by Euler's formula for positive even n, also valid for n = 0:

The functional equation of ζ(s) leads to the following formula for negative integer arguments:

Structure of the denominator
The structure of the denominator of Bn for positive even n is given by the Clausen - von Staudt Theorem:

Structure of the numerator
The numerator of Bn/n for positive even n equals 1 only for n = 2,4,6,8,10,14; otherwise this numerator is a product of powers of irregular primes. Since Bn/n is a p-integer for primes p where p-1 does not divide n, the structure of the numerator of Bn is given by

The first product is a trivial factor of Bn which divides n. The second product consists only of powers of irregular primes pv which are not easy to determine for larger n.
Kummer congruences
The Kummer congruences describe the most important arithmetical properties of the Bernoulli numbers which give a modular relation between these numbers.

The two millionth Bernoulli number

More than 10 million digits were omitted in the middle of the numerator!
The 1.5 millionth Bernoulli number

More than 7.4 million digits were omitted in the middle of the numerator!
The one millionth Bernoulli number

More than 4.7 million digits were omitted in the middle of the numerator!
Program Calcbn - A multi-threaded program for computing Bernoulli numbers via Riemann zeta function

Version
Windows (32-bit) / Exe
single & multi-threaded
Linux (64-bit) / Source
single & multi-threaded
2.0
calcbn32exe.zip (150 KB)
 calcbn64src.zip (26 KB)

Calcbn can be freely used without any warranty. Calcbn depends on the GMP library, so use the latest version of GMP with possible optimization for the current hardware that is being used. The source code of Calcbn is released under the terms of the GNU Public License. Timings on an Intel Core2Duo E6850 @ 3 GHz, computation of the 100000th Bernoulli number: Windows XP (32-bit): 16.8s (one thread), 8.7s (two threads). Linux openSUSE 11.0 (64-bit): 11.3s (one thread), 5.7s (two threads).
Factorization of numerators
Factorization of numerators of Bernoulli numbers with index 2 to 10000. Computed prime factors are less than one million: factors10t.txt (111 KB).
Irregular pairs of higher order
The irregular pairs of higher order describe the first appearance of higher powers of irregular prime factors of Bn/n. An irregular pair (p,n) of order r has the property that pr divides Bn/n with n < (p-1)pr-1. Note that n is always an even positive integer. For r = 1 this gives the usual definition of irregular pairs; note that the condition p divides Bn/n is then equal to p divides Bn. There exists a criterion to check whether the sequence of irregular pairs of higher order is unique. It has been proven for all irregular primes below 12 million that there are only unique sequences. Writing sequences p-adically these pairs of higher order provide an approximation of a uniquely existing zero of the p-adic zeta function associated with an irregular pair. For definition and properties see [R3]. Example:

Irregular pairs of higher order can be effectively and easily computed using Bernoulli numbers with small indices. By this means one can even predict the very large index of the first occurence of the power 3737 as listed above. Table of irregular pairs of order 10 for irregular primes below 1000: irrpairord.txt

A conjectural structural formula for the Bernoulli numbers
Assuming that all sequences of irregular pairs of higher order are unique, resp. each p-adic zeta function associated with an irregular pair (p,l) has a unique simple zero χ (p,l), one can describe the structure of divided Bernoulli numbers, resp. the value of the Riemann zeta function at negative odd integer arguments, as follows:

Under the proposed assumption, one can give some interpretation of the formula above. The denominator can be described by poles (always lying at 0) and the numerator by zeros of p-adic zeta functions measuring the distance to them using the p-adic metric induced by the standard ultrametric absolute value | |p. Note that this formula is valid for all irregular primes below 12 million. For detailed statements see [R3]. Equivalently, the formula conjecturally states for the Bernoulli numbers that

If there should exist a sequence of irregular pairs of higher order that is not unique, – the so-called singular case –, then the formulas given above remain valid. An additional product has to be added for those irregular pairs which belong to the singular case. This product can be described by trees of irregular pairs of higher order, which is much more complicated. Since there is not known any example of the singular case, the conjectural formulas are presented here in a simple form.
A conjectural structural formula for the Euler numbers
Similarly, for the Euler numbers En, n > 0 and even, one can state also a conjectural formula:

Here (p,l) are irregular pairs associated with the Euler numbers and the ξ (p,l)  are certain zeros of  p-adic L-functions.
Connections with class numbers of imaginary quadratic fields
Let h(d) denote the class number of the imaginary quadratic field Q(√d) of discriminant d < -4. For primes p > 3 one has the connections with Bernoulli and Euler numbers due to Carlitz [R1]:

Since h(-p) < p and h(-4p) < p for these cases, this shows that p cannot divide special Bernoulli and Euler numbers. That means for Bernoulli numbers that an irregular pair (p,(p+1)/2) for p ≡ 3 (mod 4) cannot exist.
Asymptotic formula
The product of Bernoulli numbers is described by the following asymptotic formula, see [R4],

with an asymptotic constant C2 = 4.855096646522... which is given by

where C1 = 1.8210174514992... is the product over all values of the Riemann zeta function at even positive integers and A = 1.2824271291... is the Glaisher-Kinkelin constant:

Links & References
  • [R1] L. Carlitz, The class number of an imaginary quadratic field, Comment. Math. Helv. 27 (1953), 338-345.
  • [R2] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, GTM, vol. 84, Springer-Verlag, 2nd edition, 1990.
  • [R3] B. C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007), no. 257, 405-441.
  • [R4] B. C. Kellner, On Asymptotic Constants Related to Products of Bernoulli Numbers and Factorials, Integers 9 (2009), 83-106.
Created by Bernd C. Kellner, Göttingen, Germany.
For questions or suggestions: bk (at) bernoulli.org
Last updated: Apr. 16, 2009Ref: http://www.bernoulli.org/
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