The Bernoulli numbers are a sequence of rational numbers that appear whenever number theory touches analysis. They show up in sums of powers, in the values of the zeta function, in the Taylor series of trigonometric and hyperbolic functions, and in the topology of manifolds.

Definition by generating function

The Bernoulli numbers $B_n$ are defined by:

$$\frac{z}{e^z - 1} = \sum_{n=0}^{\infty} B_n \frac{z^n}{n!}$$

This is a compact encoding: expand the left side as a power series and read off the coefficients.

The first few values

A striking pattern: all odd-indexed Bernoulli numbers are zero, except $B_1 = -1/2$. The even-indexed ones are nonzero, alternate in sign, and grow rapidly in absolute value.

Connection to sums of powers

The Bernoulli numbers were discovered by Jakob Bernoulli in pursuit of a general formula for $S_k(n) = 1^k + 2^k + \cdots + n^k$:

$$S_k(n) = \frac{1}{k+1} \sum_{j=0}^{k} \binom{k+1}{j} B_j \, n^{k+1-j}$$

With this formula and the Bernoulli numbers, you can compute the sum of any power. $B_0 = 1$ and $B_1 = -1/2$ are enough for $S_1(n) = n(n+1)/2$. You need up to $B_2 = 1/6$ for $S_2$, up to $B_4$ for $S_4$, and so on.

Bernoulli was so pleased with this result that he computed $S_{10}(1000)$ — the sum of the tenth powers of the first thousand numbers — to demonstrate the formula’s power.

Connection to the zeta function

Euler discovered that the Bernoulli numbers give exact values of the Riemann zeta function at positive even integers:

$$\zeta(2m) = \frac{|B_{2m}|}{2 \cdot (2m)!} (2\pi)^{2m}$$

This is why $\zeta(2) = \pi^2/6$ (from $B_2 = 1/6$), $\zeta(4) = \pi^4/90$ (from $B_4 = -1/30$), and so on. The Bernoulli numbers are the bridge between finite combinatorics (sums of powers) and infinite analysis (zeta values).

The appearance of $691$ in $B_{12} = -691/2730$ is famous — this prime shows up in unexpected places throughout number theory, including modular forms (it divides the numerator of $\zeta(12)/\pi^{12}$) and the theory of $p$-adic L-functions.

Connection to trigonometric functions

The Bernoulli numbers appear in several Taylor series:

$$\frac{z}{e^z - 1} = \sum B_n \frac{z^n}{n!} \qquad \text{(definition)}$$

$$z \cot z = \sum_{n=0}^{\infty} \frac{(-1)^n 4^n B_{2n}}{(2n)!} z^{2n}$$

$$\frac{z}{\tanh z} = \sum_{n=0}^{\infty} \frac{4^n B_{2n}}{(2n)!} z^{2n}$$

The fact that $B_{2n+1} = 0$ for $n \geq 1$ is reflected in these being even functions.

Computation

Ada Lovelace’s famous 1843 program — often called the first computer program — was an algorithm for computing Bernoulli numbers on Charles Babbage’s Analytical Engine. The Bernoulli numbers are the first thing anyone tried to compute automatically.

Today, the fastest algorithms use the connection to zeta values and can compute millions of Bernoulli numbers. But the conceptual picture remains: a single sequence that bridges finite sums, infinite series, and the deep structure of the integers.