the prime number theorem - a proof outline
This page is still in development, so it is debatable whether the current
contents could be correctly described as a 'proof outline'. For now the idea
is to provide a clear description of all of the key elements in the conventional
proof. This is aimed at, for example, physicists who might have a
recently developed interest in number theory, so that they can quickly get
a basic understanding of why ~ x/log
x and how/why the nontrivial zeros of the zeta function are involved.
The Prime Number Theorem (PNT) is an asymptotic law governing the
prime counting function . It simply states . This
was initially an empirical observation, noticed by
Gauss as a teenager after he had been studying a list of prime numbers.
It's not unreasonable to imagine that there might be some other way of
counting primes, involving logarithmic weighting, so that the new counting
function is asymptotic to the simpler function x.
This is indeed the case. If we define to be the function which counts
each natural number n less than or equal to x with the
= log(pk)/k = log(p)
if n = pk and zero otherwise,
then empirically we find
Chebyshev's psi function which, importantly, counts
both primes and their powers.
It's not difficult to show the equivalence of the statements
Hence to prove the PNT, we need only prove .
Taking the derivative of the logarithm of the
Riemann zeta function (using its infinite product expansion), it follows
where is the
function. The right-hand side is an example of a
Recall that .
(a general result from the theory of
we get the following formula involving a contour integral along the
vertical line Re[s] = c in the complex plane:
The function is a very slightly
altered version of the function .
At points of discontinuity, it has a value halfway between its limits on
We can now evaluate the path integral in the above equation using a limit
of expanding rectangular contours, the right-hand sides of which lie on the
vertical line Re[s] = c.
All singularities of the integrand function will eventually be enclosed in
these contours, so the theory of residues can be applied.
In this way we are able to obtain an expression
for in terms of the (x-dependent)
residues of the function .
This function clearly has singularities
when s = 0 and when = 0.
Recall that has both
trivial and nontrivial zeros.
Also, there is a singularity at s = 1, which is a simple pole of the
zeta function itself.
This then produces the explicit formula
which is a sum of x-dependent residues, and is asymptotic to
Here the are
the nontrivial zeros of the Riemann zeta function.
Now we need only make use of the properties and locations of the nontrivial
zeros in order to deal with the term
and show that it is well-enough behaved so that and
hence as required.
In fact, it is sufficient to
demonstrate that the nontrivial zeros all lie in the interior of the critical
strip, that is, that has no zeros
on the line Re[s] = 1.
important note: The PNT states that ~ Li(x), whereas the Riemann Hypothesis is equivalent to
= Li(x) +
'error term' in the PNT is intimately linked to the locations of the
nontrivial zeros. The error term can be improved by restricting these
locations - that is, by showing ever-larger regions of the critical strip
to be zero-free. The Riemann Hypothesis proposes the ultimate restriction
of the nontrivial zeros, or equivalently the maximal zero-free region of the
G.H. Hardy and M. Riesz, the General Theory of Dirichlet's Series
(Cambridge University Press, 1915)
E.C. Titchmarsh, The Zeta-Function of Riemann (Cambridge University
A.E. Ingham, The Distribution of Prime Numbers (Cambridge University
T.M. Apostol, Introduction to Analytic Number Theory (Springer, 1991)