In the middle of the 1859 paper on prime numbers, Riemann writes a sentence almost in passing. Speaking of a function linked to zeta, whose real roots correspond to the zeros of zeta on the critical line, he says:

It is very probable that all the roots are real. One would of course wish for a rigorous proof of this; but I have, after some fleeting vain attempts, provisionally put aside the search for one, since it did not seem necessary for the next objective of my investigation.

— Bernhard Riemann, On the Number of Primes Less Than a Given Magnitude (1859)

In modern language, the sentence is the Riemann hypothesis:

All the non-trivial zeros of the zeta function have real part equal to $\frac{1}{2}$.

The critical line

The trivial zeros of zeta are at the negative even integers. All the others lie in the strip $0 < \sigma < 1$ and, by the functional equation, appear symmetrically with respect to the line $\sigma = \frac{1}{2}$, the critical line. The hypothesis says that they all lie exactly on it.

On the critical line, $\zeta\left(\frac{1}{2} + it\right)$ can be multiplied by a factor of modulus 1 to become a real function, $Z(t)$, which changes sign at each zero. The graph shows $Z(t)$ between $t = 10$ and $t = 60$:

The first zeros are at $t \approx 14.135$, $21.022$, $25.011$, $30.425$, $32.935$… They follow no simple pattern, but they get closer and closer together: up to height $T$ there are about $\frac{T}{2\pi}\ln\frac{T}{2\pi e}$ zeros.

Riemann’s secret calculations

Riemann published no calculation of zeros. For that reason, for decades it was thought that the phrase “very probable” was a guess. In 1932 Carl Ludwig Siegel examined the notes Riemann left in Göttingen and found in them a sophisticated asymptotic formula for computing $\zeta$ on the critical line, with which Riemann had found the first zeros. The formula, now called the Riemann–Siegel formula, is the one the figure above uses, and it is still the basis of modern computations.

What is known

  • In 1914, G. H. Hardy proved that infinitely many zeros lie on the critical line.
  • In 1942, Atle Selberg showed that a positive proportion of them lie on the line; Brian Conrey, in 1989, that more than 40% do.
  • Computers have already verified more than ten trillion zeros: all on the critical line.

None of this is a proof. The numbers involved in the theory of primes are so large that a counterexample could lie far beyond any computation.

Why it matters

If the hypothesis is true, the error in counting primes is as small as possible: $\pi(x)$ differs from $\operatorname{li}(x)$ by at most something of the order of $\sqrt{x}\,\ln x$. The primes would be distributed as regularly as their seemingly random character allows. Hundreds of theorems have already been proved in the form “if the Riemann hypothesis is true, then…”, and all of them would gain a proof at once.

Exercise: how many zeros up to 50?

Use the formula $N(T) \approx \frac{T}{2\pi}\ln\frac{T}{2\pi e}$ to estimate how many zeros there are with $0 < t < 50$, and compare with the figure.

$\frac{50}{2\pi} \approx 7.96$ and $\frac{50}{2\pi e} \approx 2.93$, whose logarithm is $\approx 1.07$. So $N(50) \approx 7.96 \times 1.07 \approx 8.5$. The full formula has an extra term $+\frac{7}{8}$, which brings it to $\approx 9.4$. In the figure there are 10 zeros up to $t = 50$: 14.1, 21.0, 25.0, 30.4, 32.9, 37.6, 40.9, 43.3, 48.0 and 49.8. The estimate is good even for small values.

Exercise: the symmetry of the zeros

The functional equation implies that, if $\rho$ is a non-trivial zero, then so is $1 - \rho$. Moreover, since $\zeta$ takes real values at real numbers, the conjugate $\bar{\rho}$ is also a zero. If $\rho = \sigma + it$ were a zero off the critical line, how many zeros would appear “for free”?

There are four zeros: $\sigma + it$, $\sigma - it$, $(1 - \sigma) - it$ and $(1 - \sigma) + it$, at the vertices of a rectangle centred on the critical line. If $\sigma = \frac{1}{2}$, the rectangle shrinks to two points, $\frac{1}{2} \pm it$. A counterexample to the hypothesis would always come with at least three others.

Sources: Bernhard Riemann, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (1859) · Carl Ludwig Siegel, “Über Riemanns Nachlaß zur analytischen Zahlentheorie”, Quellen und Studien zur Geschichte der Mathematik 2 (1932) · Enrico Bombieri, “Problems of the Millennium: The Riemann Hypothesis”, Clay Mathematics Institute (2000) · Harold M. Edwards, Riemann’s Zeta Function (1974) · Xavier Gourdon, “The 10¹³ first zeros of the Riemann Zeta function, and zeros computation at very large height” (2004).

All topics on Bernhard Riemann · ← Counting the prime numbers · The legacy of Bernhard Riemann →

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