Riemann Hypothesis

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The real part of any non-trivial zero of the Riemann zeta function is ½.

Simply: Some complex numbers - made up of ordinary numbers between 0 and 1, combined with a
multiple of the square root of -1 - when fed into the zeta function produce the result zero. Do the infinity
of such zeroes when graphed all lie on the same critical vertical line?

Discipline: Number theory. Hundreds of results in number theory now begin, “If the Riemann
hypothesis is true, then...”

Originator: Georg Friedrich Bernhard Riemann, 1826-1866; German mathematician.

Incentive: $US1million, one of the seven Millennium Prize Problems of the USA-based Clay
Mathematics Institute.

Attempted proofs: Supercomputer numbercrunching has shown the hypothesis to be true
for more than the first billion zeros. However, the hypothesis would be wrong if only one of the
infinite results involved lies off the critical line. Several purported proofs have yet to be examined.

Is related to: Prime numbers. When the number of primes existing below a given number is plotted on a
graph, it produces a smooth curve with small wiggles - the wiggles are the Riemann zeros. Riemann found
that if the zeros do lie on the critical line then the maddeningly random distribution of prime numbers
is predictable.

Unusual aspect: May be solved via similarities with quantum mechanics. French mathematician Alain
Connes has constructed a quantum state space of infinite dimensions from the known prime numbers.
In the first dimension, measurements are made with 2-adic geometry, which pulls together even numbers.
The second dimension uses 3-adic geometry, the third 5-adic geometry and so on. Connes proved that
the system has energy levels corresponding to all the Riemann zeros that lie on the vertical line, but he still
has to prove that there are no zeros unaccounted for by these energy levels.

Could lead to: An efficient way of deciding whether a certain very large number is a prime. Mathematics
based on the Riemann zeta function could predict the behaviour of chaotic quantum systems, such as
the scattering of high energy levels in atoms and molecules, and the way in which sound and light
waves bounce around.

NZIMA connection: Marcus du Sautoy, author of The Music of the Primes, which describes the hypothesis
and its implications for a lay audience, will be visiting New Zealand as an NZIMA Maclaurin Fellow in
February and March this year.

Published in IMAges 2 - March 2007