Bernoulli Numbers
by aoum, Mar 17, 2025, 11:34 PM
Bernoulli Numbers: A Deep Dive into a Fascinating Sequence
The Bernoulli numbers are a sequence of rational numbers that play a central role in number theory, analysis, and combinatorics. They appear in many fundamental formulas, including the closed-form expression for the sum of powers of integers, the Euler-Maclaurin formula, and special values of the Riemann zeta function.
1. Definition of Bernoulli Numbers
The Bernoulli numbers, denoted by
, are defined through the following generating function:
![\[
\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}.
\]](//latex.artofproblemsolving.com/6/c/3/6c3d694e732e7579b4a30e39b5423bdac308e5b3.png)
This compact form provides a way to derive all Bernoulli numbers systematically.
First few Bernoulli numbers:
![\[
B_0 = 1, \quad B_1 = -\frac{1}{2}, \quad B_2 = \frac{1}{6}, \quad B_3 = 0, \quad B_4 = -\frac{1}{30}, \quad B_5 = 0, \quad B_6 = \frac{1}{42}, \dots
\]](//latex.artofproblemsolving.com/f/1/2/f128a7f07de2ed81187e1f91c8f53d32b5a72193.png)
Some important properties:
2. Recursive Formula for Bernoulli Numbers
Bernoulli numbers can also be computed using the following recursive relation:
![\[
B_0 = 1, \quad \sum_{k=0}^{n} \binom{n + 1}{k} B_k = 0 \quad \text{for all } n \geq 1.
\]](//latex.artofproblemsolving.com/8/5/c/85cc45f77ce3445861918f7e3e6aeed623cf8d8a.png)
This recurrence relation gives an efficient way to calculate the Bernoulli numbers.
3. Bernoulli Numbers and the Sum of Powers
One of the most famous applications of Bernoulli numbers is the formula for the sum of powers of the first
integers:
![\[
\sum_{k=1}^{n} k^m = \frac{1}{m + 1} \sum_{j=0}^{m} \binom{m + 1}{j} B_j n^{m + 1 - j}.
\]](//latex.artofproblemsolving.com/d/1/b/d1b832112106fe95ad1512f61433c652cf2a5fae.png)
For example:
![\[
\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6},
\]](//latex.artofproblemsolving.com/c/b/c/cbc048ad0db1f535b6864b7070981253d64737da.png)
which corresponds to
.
4. Bernoulli Numbers and the Euler-Maclaurin Formula
The Euler-Maclaurin formula connects Bernoulli numbers to the approximation of sums by integrals:
![\[
\sum_{k=a}^{b} f(k) = \int_a^b f(x) \, dx + \frac{f(a) + f(b)}{2} + \sum_{n=1}^{\infty} \frac{B_{2n}}{(2n)!} f^{(2n-1)}(x) \bigg|_a^b.
\]](//latex.artofproblemsolving.com/d/b/6/db6f2c98a9b3ce97c0c3faaa28adc6b9c6c441fa.png)
This formula is vital for approximating sums and evaluating asymptotic series.
5. Bernoulli Numbers and the Riemann Zeta Function
A profound connection exists between Bernoulli numbers and the values of the Riemann zeta function at even positive integers:
![\[
\zeta(2n) = (-1)^{n+1} \frac{(2\pi)^{2n} B_{2n}}{2(2n)!}.
\]](//latex.artofproblemsolving.com/7/7/3/77318e5aa4536b536d7a7f233595e8b7e5a486d3.png)
For example:
![\[
\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6},
\]](//latex.artofproblemsolving.com/e/f/5/ef5b1d1eac690534d787a9afb389a6b82c3e6c0b.png)
which corresponds to.
6. Explicit Formula for Bernoulli Numbers
There is also an explicit formula known as Akiyama-Tanigawa representation:
![\[
B_n = \sum_{k=0}^{n} (-1)^k \binom{n}{k} \frac{k^n}{k + 1}.
\]](//latex.artofproblemsolving.com/2/5/b/25b09a50ba82f21e12a2ca3770b5b8b9982acfc8.png)
This formula provides a direct method to compute Bernoulli numbers without using recursion.
7. Bernoulli Polynomials
The Bernoulli numbers generalize to Bernoulli polynomials, defined by the generating function:
![\[
\frac{te^{xt}}{e^t - 1} = \sum_{n=0}^{\infty} B_n(x) \frac{t^n}{n!}.
\]](//latex.artofproblemsolving.com/8/a/2/8a208f9b729efcc59d531dcb3c6a36030c028b35.png)
The relationship between the Bernoulli numbers and Bernoulli polynomials is:
![\[
B_n(0) = B_n.
\]](//latex.artofproblemsolving.com/3/8/7/387fa494cac66409a64b6c5f04169762eb95333b.png)
For example:
![\[
B_0(x) = 1, \quad B_1(x) = x - \frac{1}{2}, \quad B_2(x) = x^2 - x + \frac{1}{6}.
\]](//latex.artofproblemsolving.com/f/1/f/f1f4819cfaad240bdadea163d54a23941c7e3eaf.png)
8. Asymptotics of Bernoulli Numbers
For large, Bernoulli numbers grow very quickly in magnitude. Asymptotically, we have:
![\[
|B_{2n}| \approx 4 \sqrt{\pi n} \left( \frac{n}{\pi e} \right)^{2n}.
\]](//latex.artofproblemsolving.com/5/e/8/5e840f6b92b87b61d274ea2941cf2eff5d409cd7.png)
This growth rate reflects their rapid increase and irregular pattern.
9. Applications of Bernoulli Numbers
Bernoulli numbers appear in many mathematical areas, including:
10. Interesting Properties of Bernoulli Numbers
Some fascinating properties include:
11. Conclusion
Bernoulli numbers are a cornerstone of mathematical theory, connecting various disciplines through their deep and elegant properties. Their appearances in sums of powers, special functions, and zeta values make them indispensable in both pure and applied mathematics.
References
The Bernoulli numbers are a sequence of rational numbers that play a central role in number theory, analysis, and combinatorics. They appear in many fundamental formulas, including the closed-form expression for the sum of powers of integers, the Euler-Maclaurin formula, and special values of the Riemann zeta function.
A page from Seki Takakazu's Katsuyō Sanpō (1712), tabulating binomial coefficients and Bernoulli numbers
1. Definition of Bernoulli Numbers
The Bernoulli numbers, denoted by
, are defined through the following generating function:![\[
\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}.
\]](http://latex.artofproblemsolving.com/6/c/3/6c3d694e732e7579b4a30e39b5423bdac308e5b3.png)
This compact form provides a way to derive all Bernoulli numbers systematically.
First few Bernoulli numbers:
![\[
B_0 = 1, \quad B_1 = -\frac{1}{2}, \quad B_2 = \frac{1}{6}, \quad B_3 = 0, \quad B_4 = -\frac{1}{30}, \quad B_5 = 0, \quad B_6 = \frac{1}{42}, \dots
\]](http://latex.artofproblemsolving.com/f/1/2/f128a7f07de2ed81187e1f91c8f53d32b5a72193.png)
Some important properties:
- All odd Bernoulli numbers (except
) are zero: 
. - The nonzero Bernoulli numbers alternate in sign.
- The denominator of
(in reduced form) always contains the prime numbers dividing 
.
)
.
(in reduced form) always contains the prime numbers dividing 
.2. Recursive Formula for Bernoulli Numbers
Bernoulli numbers can also be computed using the following recursive relation:
![\[
B_0 = 1, \quad \sum_{k=0}^{n} \binom{n + 1}{k} B_k = 0 \quad \text{for all } n \geq 1.
\]](http://latex.artofproblemsolving.com/8/5/c/85cc45f77ce3445861918f7e3e6aeed623cf8d8a.png)
This recurrence relation gives an efficient way to calculate the Bernoulli numbers.
3. Bernoulli Numbers and the Sum of Powers
One of the most famous applications of Bernoulli numbers is the formula for the sum of powers of the first
integers:![\[
\sum_{k=1}^{n} k^m = \frac{1}{m + 1} \sum_{j=0}^{m} \binom{m + 1}{j} B_j n^{m + 1 - j}.
\]](http://latex.artofproblemsolving.com/d/1/b/d1b832112106fe95ad1512f61433c652cf2a5fae.png)
For example:
![\[
\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6},
\]](http://latex.artofproblemsolving.com/c/b/c/cbc048ad0db1f535b6864b7070981253d64737da.png)
which corresponds to
.4. Bernoulli Numbers and the Euler-Maclaurin Formula
The Euler-Maclaurin formula connects Bernoulli numbers to the approximation of sums by integrals:
![\[
\sum_{k=a}^{b} f(k) = \int_a^b f(x) \, dx + \frac{f(a) + f(b)}{2} + \sum_{n=1}^{\infty} \frac{B_{2n}}{(2n)!} f^{(2n-1)}(x) \bigg|_a^b.
\]](http://latex.artofproblemsolving.com/d/b/6/db6f2c98a9b3ce97c0c3faaa28adc6b9c6c441fa.png)
This formula is vital for approximating sums and evaluating asymptotic series.
5. Bernoulli Numbers and the Riemann Zeta Function
A profound connection exists between Bernoulli numbers and the values of the Riemann zeta function at even positive integers:
![\[
\zeta(2n) = (-1)^{n+1} \frac{(2\pi)^{2n} B_{2n}}{2(2n)!}.
\]](http://latex.artofproblemsolving.com/7/7/3/77318e5aa4536b536d7a7f233595e8b7e5a486d3.png)
For example:
![\[
\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6},
\]](http://latex.artofproblemsolving.com/e/f/5/ef5b1d1eac690534d787a9afb389a6b82c3e6c0b.png)
which corresponds to
.6. Explicit Formula for Bernoulli Numbers
There is also an explicit formula known as Akiyama-Tanigawa representation:
![\[
B_n = \sum_{k=0}^{n} (-1)^k \binom{n}{k} \frac{k^n}{k + 1}.
\]](http://latex.artofproblemsolving.com/2/5/b/25b09a50ba82f21e12a2ca3770b5b8b9982acfc8.png)
This formula provides a direct method to compute Bernoulli numbers without using recursion.
7. Bernoulli Polynomials
The Bernoulli numbers generalize to Bernoulli polynomials, defined by the generating function:
![\[
\frac{te^{xt}}{e^t - 1} = \sum_{n=0}^{\infty} B_n(x) \frac{t^n}{n!}.
\]](http://latex.artofproblemsolving.com/8/a/2/8a208f9b729efcc59d531dcb3c6a36030c028b35.png)
The relationship between the Bernoulli numbers and Bernoulli polynomials is:
![\[
B_n(0) = B_n.
\]](http://latex.artofproblemsolving.com/3/8/7/387fa494cac66409a64b6c5f04169762eb95333b.png)
For example:
![\[
B_0(x) = 1, \quad B_1(x) = x - \frac{1}{2}, \quad B_2(x) = x^2 - x + \frac{1}{6}.
\]](http://latex.artofproblemsolving.com/f/1/f/f1f4819cfaad240bdadea163d54a23941c7e3eaf.png)
8. Asymptotics of Bernoulli Numbers
For large
, Bernoulli numbers grow very quickly in magnitude. Asymptotically, we have:![\[
|B_{2n}| \approx 4 \sqrt{\pi n} \left( \frac{n}{\pi e} \right)^{2n}.
\]](http://latex.artofproblemsolving.com/5/e/8/5e840f6b92b87b61d274ea2941cf2eff5d409cd7.png)
This growth rate reflects their rapid increase and irregular pattern.
9. Applications of Bernoulli Numbers
Bernoulli numbers appear in many mathematical areas, including:
- Number Theory: Understanding congruences, modular forms, and the Riemann zeta function.
- Combinatorics: Calculating sums of powers and advanced recurrence relations.
- Analysis: Euler-Maclaurin formula and asymptotic approximations.
- Topology: Invariants of certain manifolds through zeta-function regularization.
10. Interesting Properties of Bernoulli Numbers
Some fascinating properties include:
- Kummer’s Congruence: For any prime
, if 
and are positive integers such that 
, then:
![\[
\frac{B_m}{m} \equiv \frac{B_n}{n} \mod p.
\]](//latex.artofproblemsolving.com/8/d/0/8d0460e28d02b2ba763e7fb77e6b28187abe1d03.png)
- Irregular Primes: A prime
is called irregular if it divides the numerator of some . For example, 37 is an irregular prime because it divides 
.
,
and 
,![\[
\frac{B_m}{m} \equiv \frac{B_n}{n} \mod p.
\]](http://latex.artofproblemsolving.com/8/d/0/8d0460e28d02b2ba763e7fb77e6b28187abe1d03.png)
is called irregular if it divides the numerator of some 
.11. Conclusion
Bernoulli numbers are a cornerstone of mathematical theory, connecting various disciplines through their deep and elegant properties. Their appearances in sums of powers, special functions, and zeta values make them indispensable in both pure and applied mathematics.
References
- Wikipedia: Bernoulli Numbers
- Apostol, T. Introduction to Analytic Number Theory.
- Ireland, K., Rosen, M. A Classical Introduction to Modern Number Theory.
March 2025