I’ve read 3 books on
this topic: The Music of the Primes
by Marcus du Sautoy, Prime Obsession
by John Derbyshire and Stalking the
Riemann Hypothesis by Dan Rockmore (and I originally read them in that
order). They are all worthy of recommendation, but only John Derbyshire makes a
truly valiant attempt to explain the mathematics behind the ‘Hypothesis’ (for
laypeople) so it’s his book that I studied most closely.
Now it’s impossible
for me to provide an explanation for 2 reasons: one, I’m not mathematically
equipped to do it; and two, this is a blog and not a book. So my intention is
to try and instill some of the wonder that Riemann’s extraordinary
gravity-defying intuitive leap passes onto those who can faintly grasp its
mathematical ramifications (like myself).
In 1859 (the same year
that Darwin published The Origin of the
Species), a young Bernhard Riemann (aged 32) presented a paper to the
Berlin Academy as part of his acceptance as a ‘corresponding member’, titled
“On the Number of Prime Numbers Less Than a Given Quantity”. The paper contains
a formula that provides a definitive number called π (not to be confused with
pi, the well-known transcendental number). In fact, I noticed that Derbyshire uses π(x) as a function in an attempt to make a distinction. As Derbyshire points out, it’s
a demonstration of the limitations arising from the use of the Greek alphabet
to provide mathematical symbols – they double-up. So π(x) is the number of primes
to be found below any positive Real number. Real numbers include rational
numbers, irrationals and transcendental numbers, as well as integers. The
formula is complex and its explication requires a convoluted journey into the
realm of complex algebra, logarithms and calculus.
Eratosthenes was one
of the librarians at the famous Alexandria Library, around 230 BC and roughly 70
years after Euclid. He famously measured the circumference of the Earth to
within 2% of its current figure (see Wikipedia) using the sun and some basic
geometry. But he also came up with the first recorded method for finding primes
known as Eratosthenes’ Sieve. It’s so simple that it’s obvious once explained: leaving
the number 1, take the first natural number (or integer) which is 2, then
delete all numbers that are multiples of 2, which are all the even numbers.
Then take the next number, 3, and delete all its multiples. The next number
left standing is 5, and one just repeats the process over whatever group of
numbers one is examining (like 100, for example) until you are left with all
the primes less than 100. With truly gigantic numbers there are other methods,
especially now we have computers that can grind out algorithms, but
Eratosthenes demonstrates that scholars were fascinated by primes even in antiquity.
Euclid famously came
up with a simple proof to show that there are an infinite number of primes,
which, on the surface, seems a remarkable feat, considering it’s impossible to
count to infinity. But it’s so simple that Stephen Fry was even able to explain
it on his TV programme, QI. Assume
you have found the biggest prime, then take all the primes up to and including
that prime and multiply them all together. Then add 1. Obviously none of the
primes you know can be factors of this number as they would all give a
remainder of 1. Therefore the number is either a prime or can be factored by a
prime that is higher than the ones you already know. Either way, there will
always be a higher prime, no matter which one you select, so there must be an
infinite number of primes.
The thing about
primes, that has fascinated mathematicians for eons, is that there appears to
be no rhyme or reason to their distribution, except they get thinner - further
apart as one goes to higher numbers. But even this is not strictly correct
because there appears to be an infinite number of twin primes, 2 primes
separated by a non-prime (which must be even for obvious reasons).
Back to Riemann’s
paper and its 150 year old legacy. Entailed in his formula is a formulation of
the Zeta function. Richard Elwes provides a relatively succinct exposition in
his encyclopaedic MATHS 1001, and I’m not even going to attempt to write it
down here. The point is that the
Zeta function gives complex roots to infinity. Most people know what a
quadratic root is from high school maths. If you take the graph of a parabola
like y = ax2 + bx + c, then it crosses the x axis where y = 0. It
can cross the x axis in 2 places, or just touch it in 1 place or not cross it
at all. The values of x that gives us a 0 value of y are called the roots of
the equation. As a polynomial goes up in degree so does its number of roots. So
a quadratic equation gives us 2 roots maximum but a polynomial with degree 3
(includes x3) will give us 3 roots and so on. Going back to the
parabola, in the case where we don’t get any roots at all, it’s because we are
trying to find square roots of negative numbers. However, if we use i (√-1), we get complex roots in the form of a +
ib. (For a basic explanation see my Apr.12 post on imaginary numbers.) A
trigonometric equation like sinθ can give us an infinite number of zeros and so
can the Zeta function.
If you didn’t follow
that, don’t worry, the important point is that Riemann’s Hypothesis says that all
the complex zeros of the Zeta function (to infinity) have Real part ½. So they
are all of the form ½ + ib. Riemann wasn’t able to provide a proof for this and
neither have the best mathematical minds since. The critical point is that if
his Hypothesis is correct then so is his formula for finding an exact number of
primes to any given number.
In the 150 years
since, Riemann’s Hypothesis has found its way into many fields of mathematics,
including Hermitian matrices, which has implications for quantum mechanics. The
Zeta function is a formidable mathematical beast to the uninitiated, and its relationship to the distribution of the primes was first intimated by Euler.
Riemann’s genius was to introduce complex numbers, then make the convoluted
mental journey to demonstrate their pivotal role in providing an exact result. Even then, his fundamental conjecture was effectively based on
a hunch. At the time he presented his paper, he had only calculated the first 3
non-trivial zeros (non-trivial means complex in this context) and computers
have calculated them in the trillions since, yet we still have no proof. It’s known that
they become chaotic at extremely high numbers (beyond the number of atoms in
the universe) so it’s by no means certain that Reimann’s hypothesis is correct.
It would be a huge
disappointment to most mathematicians if either a proof was found to falsify it
or an exception was found through brute computation. Riemann gave us a formula that
gives us an accurate count of the primes (Derbyshire gives a worked example up to 1 million) that’s dependent on the Hypothesis being correct to specified values.
It’s hard to imagine that this formula suddenly fails at some extremely high
number that’s currently beyond our ken, yet it can’t be ruled out.
Marcus du Sautoy, in The Music of the Primes, contemplates
the Riemann hypothesis in the context of Godel’s Incompleteness Theorem, which
is germane to the entire edifice of mathematics. The primes have a history of
providing hard-to-prove conjectures. Along with Riemann’s hypothesis, there is
the twin prime conjecture I mentioned earlier and Goldbach’s conjecture, which
states that every even number greater than 2 is the sum of 2 primes. These
conjectures are also practical demonstrations of Turing’s halting problem
concerning computers. If they are correct, a computer algorithm set to finding
them may never stop, yet we can’t determine in advance whether it will or not,
otherwise we’d know in advance if it was true or not.
As du Sautoy points
out, a corollary to Godel’s theorem is that there are limits to the proofs from
any axioms we know at any time. In essence, there may be mathematical truths
that the axioms cannot cover. The solution is to expand the axioms. In other
words, we need to expand the foundations of our mathematics to extend our
knowledge at its stratospheric limits. Du Sautoy speculates that the Riemann
Hypothesis, along with these other examples, may be Godel’s Incompleteness Theorem
in action.
Exploring the Reimann
Hypothesis, even at the rudimentary level that I can manage, reinforces my
philosophical Platonist view of mathematics. These truths exist independently
of our investigations. There are an infinity of these Zeta zeros (we know that
much) the same as there are an infinity of primes, which means there will
always exist mathematical entities that we can’t possibly know. But aside from
that obvious fact, the relationship that exists between apparently obscure
objects like Zeta zeros and the distribution of prime numbers is a wonder. Godel’s
Theorem implies that no matter how much we learn, there will always be
mathematical wonders beyond our ken.
Addendum: This is a reasonably easy-to-follow description of Riemann's famous Zeta function, plus lots more.
Addendum: This is a reasonably easy-to-follow description of Riemann's famous Zeta function, plus lots more.
