Everything about The Riemann Hypothesis totally explained
The
Riemann hypothesis (also called the
Riemann zeta-hypothesis), first formulated by
Bernhard Riemann in 1859, is one of the most famous and important unsolved problems in
mathematics. It has been an open question for almost 150 years, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it's more attractive to professionals in the field than to amateurs.
The Riemann hypothesis (RH) is a
conjecture about the distribution of the
zeros of the
Riemann zeta-function ζ(
s). The Riemann zeta-function is defined for all
complex numbers
s ≠ 1. It has zeros at the negative even integers (for example at
s = −2,
s = −4,
s = −6, ...). These are called the
trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that:
» The real part of any non-trivial zero of the Riemann zeta function is ½.
Thus the non-trivial zeros should lie on the so-called
critical line, ½ +
it, where
t is a
real number and
i is the
imaginary unit. The Riemann
zeta-function along the critical line is sometimes studied in terms of the
Z-function, whose real zeros correspond to the zeros of the zeta-function on the critical line.
The Riemann hypothesis is one of the most important open problems of contemporary mathematics, mainly because a large number of deep and important other results have been proven under the condition that it holds. Most mathematicians believe the Riemann hypothesis to be true. A $1,000,000 prize has been offered by the
Clay Mathematics Institute for the first correct proof.
History
Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper
On the Number of Primes Less Than a Given Magnitude, but as it wasn't essential to his central purpose in that paper, he didn't attempt a proof. Riemann knew that the non-trivial zeros of the zeta-function were symmetrically distributed about the line
s = ½ +
it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(
s) ≤ 1.
In 1896,
Hadamard and
de la Vallée-Poussin independently proved that no zeros could lie on the line Re(
s) = 1. Together with the other properties of non-trivial zeros proved by Riemann, this showed that all non-trivial zeros must lie in the interior of the critical strip 0 < Re(
s) < 1. This was a key step in the first complete proofs of the
prime number theorem.
In 1900,
Hilbert included the Riemann hypothesis in his famous list of
23 unsolved problems — it's part of Problem 8 in Hilbert's list, along with the
Goldbach conjecture. When asked what he'd do if awakened after having slept for five hundred years, Hilbert famously said his first question would be whether the Riemann hypothesis had been proven (Derbyshire 2003:197; Sabbagh 2003:69; Bollobas 1986:16). The Riemann Hypothesis is the only one of Hilbert's problems on the
Clay Mathematics Institute Millennium Prize Problems.
In 1914,
Hardy proved that an infinite number of zeros lie on the critical line Re(
s) = ½. However, it was still possible that an infinite number (and possibly the majority) of non-trivial zeros could lie elsewhere in the critical strip.
Later work by Hardy and
Littlewood in 1921 and by Selberg in 1942 gave estimates for the average density of zeros on the critical line.
Recent work has focused on the explicit calculation of the locations of large numbers of zeros (in the hope of finding a counterexample) and placing upper bounds on the proportion of zeros that can lie away from the critical line (in the hope of reducing this to zero).
The fractal structure of the Riemann zeta zeros has been studied using
Rescaled Range Analysis.
The self-similarity of the zero distributions is quite remarkable,
and is characterized by a large fractal dimension of 1.9.
The Riemann hypothesis and primes
The traditional formulation of the Riemann hypothesis obscures somewhat the true importance of the conjecture. The zeta-function has a deep connection to the distribution of
prime numbers.
Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the following considerable strengthening of the
prime number theorem: for every ε > 0, we have
» and numerical evidence doesn't suggest it
can grow nearly as fast as the Riemann hypothesis seems to allow, much less
as fast as the best that can at present be shown without it.
Attempted proofs of the Riemann hypothesis
Several teams of mathematicians have addressed the Riemann hypothesis over decades, and a few purported proofs go unverified
as of 2007. However, these have been received with skepticism by the mathematical community, and professionals at large don't believe them to be true. Matthew R. Watkins from the
University of Exeter has a compilation of such claims (serious and ludicrous alike), and a few others may be found in the
arXiv database.
Possible connection with operator theory
It has long been speculated that the
correct way to derive the Riemann hypothesis has been to find a
self-adjoint operator, from the existence of which the statement on the real parts of the zeroes of ζ(
s) would follow when one applies the criterion on real
eigenvalues. This has led to many investigations, but hasn't yet proven fruitful.
The distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of
random matrices drawn from the
Gaussian unitary ensemble. This gives some support to the Hilbert–Pólya conjecture.
In 1999,
Michael Berry and Jon Keating conjectured that there's some unknown quantization
of the classical Hamiltonian
so that
»
and even more strongly, that the Riemann zeros coincide with the spectrum of the operator
. This is to be contrasted to
canonical quantization which leads to the
Heisenberg uncertainty principle and the
natural numbers as spectrum of the
quantum harmonic oscillator. The crucial point is that the Hamiltonian should be a
Hermitian operator (or more precisely closed
self adjoint operator) so that the quantisation would be a realisation of the Hilbert–Pólya program.
Searching for ζ-function zeroes
There is a long history of computational attempts to explore as many zeroes of the ζ-function as possible. One notable such attempt was
ZetaGrid, a
distributed computing project, which checked over a billion zeros a day when it was running. The project was shut down in November 2005.
As of 2006, no computational project has succeeded in finding a counterexample to the Riemann hypothesis.
In 2004,
Xavier Gourdon and
Patrick Demichel verified the Riemann hypothesis through the first ten trillion non-trivial zeros using the
Odlyzko-Schönhage algorithm.
Michael Rubinstein has made public an
algorithm
for generating the zeros.
Further Information
Get more info on 'Riemann Hypothesis'.
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