Number of Bernoulli

In Mathematical, the numbers of Bernoulli , noted $B_n \,$ , were initially studied by seeking formulas to express the naps of the type:

$\ sum_ {k=0} ^ {M-1} k^n = 0^n + 1^n + 2^n + \ cdots + {(M-1)}^n$

for various values of the entirety N .

Introduction

Such expressions are always Polynôme S in m , of degree

n+1 \,

and are called polynomials of Bernoulli . The Coefficient S of the polynomials of Bernoulli are related to the numbers of Bernoulli in the following way:

$\ sum_ {k=0} ^ {M-1} k^n = {1 \ over {n+1}} \ sum_ {k=0} ^n {n+1 \ choose {K}} B_k m^ {n+1-k}$

For example, by giving to N value 1, one obtains:

$0+1+2+ \ ldots+ (M-1) = \ frac {1} {2} (B_0 m^2+2 B_1 m^1) = \ frac {1} {2} (m^2-m) \,$

The numbers of Bernoulli were initially studied by Jacques Bernoulli, which led Abraham de Moivre to give them the name that we know today.

It is possible to calculate the numbers of Bernoulli by using the formula of following recurrence:

$\ sum_ {j=0} ^m {m+1 \ choose {J}} B_j = 0 \,$

in addition to the initial condition: $B_0 = 1 \,$ .

The numbers of Bernoulli can also be defined by using the technique of the generating functions. Their generating Fonction exponential is $\ frac {X} {e^x-1} \,$ , so that:

\ frac {X} {e^x-1} = \ sum_ {n=0} ^ {\ infin} B_n \ frac {x^n} {N!} for all X of absolute value lower than

2 \ pi \,

(the ray of convergence of this whole Series).

These definitions can be shown like equivalent by using the mathematical induction. The initial condition $B_0 = 1$ is immediate starting from the Règle of the Hospital. To obtain the recurrence, one multiplies both with dimensions equation by $e^x-1$ . Then, by using the Taylor series for the exponential Function,

$x = \ left (\ sum_ {j=1} ^ {\ infty} \ frac {x^j} {J!} \ right) \ left (\ sum_ {k=0} ^ {\ infty} \ frac {B_k x^k} {K!} \ right).$

By developing this like a Produces of Cauchy and while rearranging slightly, one obtains

$X = \ sum_ {m=0} ^ {\ infty} \ left (\ sum_ {j=0} ^ {m} {m+1 \ choose J} B_j \ right) \ frac {x^ {m+1}} {(m+1)!}.$

It is clear, starting from this last equality, that the coefficients in this series of powers satisfy the same recurrence as numbers of Bernoulli.

Sometimes, the notation $b_n \,$ is used to distinguish the numbers of Bernoulli of the numbers of Beautiful.

Values

The first numbers of Bernoulli are the following:

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One can show that $B_n = 0 \,$ when N is odd and different from 1.

Appearance of the value of $B_ {12} = \ frac {- 691} {2730} \,$ watch although the value of the numbers of Bernoulli cannot be described simply; in fact, they are primarily values of the function ζ of Riemann for negative whole values of the variable, (since $\ zeta (- N) = - B_ {n+1} (n+1) \,$ for all the positive entireties N ), and, are consequently associated with major properties of the Théorie of the numbers, and cannot hope to have commonplace formulation.

The numbers of Bernoulli also appear in the development in Taylor series of the tangent functions circular and hyperbolic, in the formula of Euler-Maclaurin like in expressions of certain values of the function zeta of Riemann.

Remarkable identities

Leonhard Euler expressed the numbers of Bernoulli in terms of function zeta of Riemann:

B_ {2k} = (- 1) ^ {k-1} \ frac {\ zeta (2k) \; 2 (2k)!} {(2 \ pi) ^ {2k}} \,

The following relations, due to Ramanujan, provide a more effective method for calculation of the numbers of Bernoulli:

$m \ equiv 0 \, \ bmod \, 6 \ qquad B_m=- \ sum_ {j=1} ^ {m/6} {m+3 \ choose {m-6j}} B_ {m-6j}$

$m \ equiv 2 \, \ bmod \, 6 \ qquad B_m=- \ sum_ {j=1} ^ {(m2) /6} {m+3 \ choose {m-6j}} B_ {m-6j}$

$m \ equiv 4 \, \ bmod \, 6 \ qquadB_m=--\ sum_ {j=1} ^ {(m-4) /6} {m+3 \ choose {m-6j}} B_ {m-6j}$

An identity of Carlitz:

$(- 1) ^m \ sum_ {r=0} ^m {m \ choose R} B_ {n+r}$

(- 1) ^n \ sum_ {s0} ^n {N \ choose S} B_ {m+s}

Arithmetic properties

It is possible to express the numbers of Bernoulli thanks to the Fonction zeta of Riemann in the following way:

B_n = - N \ zeta (1-n) \,

;

This is why the numbers of Bernoulli have major arithmetic properties, like discovered Kummer in its work on the Dernier theorem of Fermat.

The properties of Divisibilité of the numbers of Bernoulli are related on the groups of the classes of ideals of the cyclotomic Corps S by a theorem of Kummer and its reinforcement in the Théorème of Herbrand-Ribet, and to the numbers of classes of the quadratic bodies by the Congruence of Ankeny-Artin-Chowla. We have also family ties with the algebraic K-theory; if $c_n \,$ is the numerator of $\ frac {B_n} {2n} \,$ , then the order of $K_ {4n-2} (\ Bbb {Z}) \,$ is $-c_ {2n} \,$ if N is even, and $2c_ {2n} \,$ if N is odd.

The Théorème of von Staudt-Clausen is also connected to divisibility. It states this: if we add $\ frac {1} {p} \,$ with $B_n \,$ for each Prime number p such as p − 1 divides N , we obtain a Integer. This fact immediately enables us to characterize the denominators of the numbers of Bernoulli different from zero $B_n \,$ like the product of all the prime numbers p such as p − 1 divides N ; consequently, the denominators are Without square and divisible by 6.

The Conjecture of Agoh-Giuga postulates that p is a prime number if and only if $pB_ {p-1} \ equiv -1 \ MOD {p} \,$ .

P-adic continuity

A property of especially important congruence of the numbers of Bernoulli can be characterized like a property of p-adic continuity. If B , m and N is positive integers such as m and N is not divisible by $p-1 \,$ and $m \ equiv N \, \ bmod \, p^ {b-1} (p-1) \,$ , then

(1-p^ {M-1}) {B_m \ over m} \ equiv (1-p^ {n-1}) {B_n \ over N} \, \ bmod \, p^b \,

Since

B_n = - N \ zeta (1-n) \,

, this can be also written

(1-p^ {- U}) \ zeta (U) \ equiv (1-p^ {- v}) \ zeta (v) \, \ bmod \, p^b \,

where

u=1-m \,

and

v=1-n \,

, i.e. U and v are negative and nonadequate to 1 MOD p-1 . This indicates to us that the function zeta of Riemann, with

1-p^z \,

taken out of the formula of the product of Euler, is continuous for the numbers p-adic on negative integers adequate MOD p-1 , in particular

a \ not \ equiv 1 \, \ bmod \, p-1

, and thus, can be wide with a continuous function

\ zeta_p (Z) \,

for all the p-adic integers

\ mathbb {Z} _p, \,

the p-adic function zeta .

Geometrical properties

The formula of Kervaire- Milnor for the order of the cyclic Group of the classes of Diffeomorphism S of (4 N −1) - exotic spheres which limit parallélisables varieties for $n \ Ge 2$ imply the numbers of Bernoulli: if B is the numerator of $\ frac {B_ {4n}} {N} \,$ , then $2^ {2n-2} (1-2^ {2n-1}) B \,$ is the number of these exotic spheres. (The formula in the topological articles differs because the topologists use a different convention to name the numbers of Bernoulli; this article uses the convention of the Théorie of the numbers).

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