You are probably familiar with the *Riemann hypothesis*. This concerns the behaviour of the Riemann zeta function, which is defined on the complex plane by analytic continuation of the following series:

The behaviour of the zeroes of the function outside the strip is well-known: they are precisely at the even negative integers. The Riemann hypothesis states that within this strip, the only zeroes have real part exactly equal to ½. It is customary to include, at this point, an entirely gratuitous plot of the real part of the zeta function near the origin:

Whilst interesting enough for the Clay Mathematics Institute to offer a prize of USD for its resolution, the *generalised Riemann hypothesis* has far more far-reaching implications. This states that for every Dirichlet L-function, all of its zeroes within the critical strip have real part exactly equal to ½. Since the Riemann zeta function is the simplest example of a Dirichlet L-function, the ordinary Riemann hypothesis is subsumed by the generalised Riemann hypothesis.

### What is a Dirichlet L-function?

A Dirichlet L-function is specified by a positive integer *k* and a completely multiplicative function satisfying:

- χ(
*n*) = 0 if and only if gcd(*n*, *k*) ≠ 1;
**Periodicity:** χ(*n* + *k*) = χ(*n*) for all integers *n*;
**Complete multiplicativity:** χ(*m*) χ(*n*) = χ(*mn*) for all integers *m*, *n*.

Given such a *Dirichlet character* χ (there are finitely many choices, namely φ(*k*), for any given *k*), the corresponding L-function is defined as:

When *k* = 1, the unique Dirichlet character is the constant function 1, whence we recover the original Riemann hypothesis.

### Knottedness is in NP, modulo GRH

This is the title of a very interesting paper by Greg Kuperberg. It has been proved that an unknotted knot diagram can be disentangled into a simple closed loop in a number of Reidemeister moves polynomial in the number of crossings in the original diagram. It follows that unknottedness is in NP, as an optimal disentangling sequence is a polynomial-time-checkable certificate of unknottedness.

Kuperberg’s paper proves that unknottedness is in co-NP, i.e. knottedness is in NP. However, the proof relies on the unproven generalised Riemann hypothesis in a very interesting way. Specifically, if the fundamental group of the complement of the knot is non-Abelian — as is the case for any non-trivial knot — then it has a non-Abelian representation over SU(2). If the generalised Riemann hypothesis holds, it is possible to find a reasonably small prime *p* for which there is such a modulo-*p* representation; in this case, the images of the generators under this representation (expressed as 2-by-2 matrices with entries modulo *p*) form a certificate for knottedness. It then suffices to check that they do not all commute, and that they satisfy the relations of the knot group presentation.

### ab + bc + ca

Which positive integers cannot be expressed in the form *ab + bc + ca*, where *a, b, c* are positive integers? There are either 18 or 19 such numbers, the first 18 of which are enumerated in A025052:

`1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462`

There may be a 19th such number, which must necessarily be greater than 10^11, and can only exist provided the generalised Riemann hypothesis is false. In particular, Jonathan Borwein proved that the existence of this 19th number requires a Dirichlet L-function to have a Siegel zero, which would contradict GRH.

The positive integers inexpressible in this form with a, b, c **distinct** positive integers are called *idoneal numbers*, and there are either 65 or 66 of them. The 65 known examples are sequence A000926:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848

Idoneal numbers have an equivalent definition of being the positive integers D such that any number uniquely expressible in the form , with and coprime, is necessarily either a prime, a prime power, or twice a prime power.

### Deterministic Miller-Rabin

The fastest known deterministic algorithm for determining if a number is prime is AKS, which takes time . This is, unfortunately, rather slow.

The *Miller-Rabin probabilistic test* tells you whether a number is prime in time . If the number is prime, it correctly claims that it is prime. If it is composite, however, it can lie with probability 1/4. In practice, running this test with many randomly-chosen initial parameters will give a result with sufficiently high probability that it is useful in practice (e.g. in cryptography).

Assuming GRH, it can be shown that applications of Miller-Rabin are sufficient to guarantee primality, and therefore that there exists a deterministic algorithm with running time , noticeably faster than AKS. The Shanks-Tonelli algorithm for finding square-roots modulo a prime will also run deterministically in polynomial time as a result of the same idea.

### Artin’s conjecture on primitive roots

This states that every integer *a* which is neither −1 nor a perfect square is a primitive root modulo infinitely many primes. In 1967, Hooley proved this conditional on the Generalised Riemann Hypothesis, establishing that the set of primes for which *a* is a primitive root has positive density (among the set of all primes).