Take the harmonic series and make the exponent a variable:

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \cdots$$

For $s = 1$, this is the harmonic series (diverges). For $s > 1$, it converges. The function $\zeta(s)$ — the Riemann zeta function — is one of the most important objects in mathematics.

Explore: partial sums of $\zeta(s)$

Drag the slider to change $s$ and watch the partial sums of $\sum 1/n^s$:

Try $s = 1$ — the harmonic series, diverging. Then slide to $s = 2$ — the Basel problem, converging to $\pi^2/6$. The transition at $s = 1$ is the pole of the zeta function.

Special values

The zeta function at positive even integers gives beautiful closed forms involving $\pi$:

The case $s = 2$ is the famous Basel problem, posed in 1650 and solved by Euler in 1734 — a result that made him famous across Europe. The general pattern is:

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

where $B_{2m}$ are the Bernoulli numbers.

The odd values ($\zeta(3), \zeta(5), \ldots$) are far more mysterious. $\zeta(3) \approx 1.2021$ (called Apéry’s constant) was proved irrational only in 1978. No closed form is known for any odd zeta value.

The Basel problem

Why is $\sum 1/n^2 = \pi^2/6$? Euler’s proof used the product formula for sine:

$$\sin x = x \prod_{k=1}^{\infty} \left(1 - \frac{x^2}{k^2\pi^2}\right)$$

Expanding this product and comparing the coefficient of $x^3$ with the Taylor series $\sin x = x - x^3/6 + \cdots$ gives:

$$\sum_{k=1}^{\infty} \frac{1}{k^2\pi^2} = \frac{1}{6}$$

Multiply by $\pi^2$: $\zeta(2) = \pi^2/6$. The appearance of $\pi$ in a sum over integers is surprising and hints at deep structure.

From sums to products

The zeta function has a second representation — as a product over primes rather than a sum over integers. This is the Euler product:

$$\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$$

This identity — which we explore in the next node — is the fundamental connection between the zeta function (analysis) and prime numbers (arithmetic). It says: the distribution of primes is encoded in the analytic properties of $\zeta$.

Beyond the series

The series $\sum 1/n^s$ only converges for $s > 1$. But through analytic continuation, the zeta function can be extended to all complex numbers $s$ except $s = 1$ (where it has a pole — the harmonic series).

The extended function has zeros at $s = -2, -4, -6, \ldots$ (the “trivial zeros”) and infinitely many other zeros in the critical strip $0 < \text{Re}(s) < 1$. The Riemann Hypothesis — the most famous unsolved problem in mathematics — conjectures that all non-trivial zeros lie on the critical line $\text{Re}(s) = 1/2$.

The location of these zeros controls the fine structure of how primes are distributed among the integers. If the Riemann Hypothesis is true, primes are distributed as regularly as they could possibly be.

The zeta function as a Dirichlet series

The zeta function is the simplest example of a Dirichlet series — a series of the form $\sum a_n / n^s$. With $a_n = 1$ for all $n$, you get $\zeta$. With other choices of $a_n$ (like the divisor function $\sigma(n)$ or characters from number theory), you get L-functions — generalisations that encode information about primes in arithmetic progressions, elliptic curves, and modular forms.