The function that bears Riemann’s name was studied earlier by Euler. For a real number $s$ greater than 1, the series

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

converges. Euler computed several of its values, such as $\zeta(2) = \frac{\pi^2}{6}$, the solution of the Basel problem.

Euler’s product

Euler’s deepest discovery, of 1737, was that the same function can be written as a product over the prime numbers:

$$\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}} = \frac{1}{1 - 2^{-s}}\cdot\frac{1}{1 - 3^{-s}}\cdot\frac{1}{1 - 5^{-s}}\cdots$$

The reason is unique factorization. Each factor is a geometric progression: $\frac{1}{1 - 2^{-s}} = 1 + \frac{1}{2^s} + \frac{1}{4^s} + \frac{1}{8^s} + \cdots$. Multiplying the factors of all the primes and expanding, each integer $n$ appears exactly once, as the product of the powers of its primes. For example, $\frac{1}{12^s}$ comes from choosing $\frac{1}{4^s}$ in the factor for 2 and $\frac{1}{3^s}$ in the factor for 3. The left-hand side speaks only of integers; the right-hand side, only of primes. The formula is a bridge between the two, and with it Euler gave a new proof that there are infinitely many primes.

The complex variable

Riemann, who had devoted his thesis to complex functions, took the decisive step in the 1859 paper: he let $s$ be a complex number, $s = \sigma + it$. The series still converges whenever the real part $\sigma$ is greater than 1, because $|n^{-s}| = n^{-\sigma}$.

But Riemann showed more: the function can be extended to the whole complex plane, with a single exception, $s = 1$, where it goes to infinity. Outside the region $\sigma > 1$ the series no longer converges, but the function goes on existing, defined by other formulas that agree with the series where it converges. This is called analytic continuation: a differentiable complex function is entirely determined by any piece of it.

The symmetry between s and 1 − s

Riemann discovered that the extended function has a remarkable symmetry, the functional equation:

$$\zeta(s) = 2^s\,\pi^{s-1}\,\sin\!\left(\frac{\pi s}{2}\right)\Gamma(1 - s)\,\zeta(1 - s),$$

where $\Gamma$ is the gamma function, which extends the factorial. The equation relates the values at $s$ to the values at $1 - s$, the point symmetric with respect to the vertical line $\sigma = \frac{1}{2}$. This line, the critical line, will be at the centre of the Riemann hypothesis.

The functional equation explains, for example, why $\zeta$ vanishes at the negative even integers, $-2, -4, -6, \ldots$: there the sine vanishes. These are the trivial zeros. All the other zeros, the non-trivial ones, lie in the strip $0 < \sigma < 1$, the critical strip.

Exercise: Euler's product for s = 2

Compute the product $\frac{1}{1 - 2^{-2}}\cdot\frac{1}{1 - 3^{-2}}\cdot\frac{1}{1 - 5^{-2}}\cdot\frac{1}{1 - 7^{-2}}$ and compare it with $\zeta(2) = \frac{\pi^2}{6} \approx 1.6449$.

The factors are $\frac{4}{3}$, $\frac{9}{8}$, $\frac{25}{24}$ and $\frac{49}{48}$. The product is $\frac{4 \cdot 9 \cdot 25 \cdot 49}{3 \cdot 8 \cdot 24 \cdot 48} = \frac{44\,100}{27\,648} \approx 1.5951$. With only the first four primes we already reach 97% of the value; the other primes make up the rest.

Exercise: the size of n to a complex power

Show that $|n^{-s}| = n^{-\sigma}$ for $s = \sigma + it$, and conclude that the series for $\zeta(s)$ converges when $\sigma > 1$.

$n^{-s} = e^{-s\ln n} = e^{-\sigma\ln n}\,e^{-it\ln n}$. The second factor has modulus 1, because it is of the form $e^{i\theta}$. So $|n^{-s}| = e^{-\sigma\ln n} = n^{-\sigma}$. The series of moduli is $\sum n^{-\sigma}$, which converges for $\sigma > 1$. Hence the complex series converges absolutely in that region.

Sources: Leonhard Euler, “Variae observationes circa series infinitas” (1737; published 1744) · Bernhard Riemann, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse”, Monatsberichte der Berliner Akademie (November 1859) · Harold M. Edwards, Riemann’s Zeta Function (1974) · John Derbyshire, Prime Obsession (2003).

All topics on Bernhard Riemann · ← The hypotheses on which geometry is founded · Counting the prime numbers →

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