The Riemann zeta function is defined to be the complex valued function given by the series
The Euler product formula (2) given above expresses the
zeta function as a product over the primes
, and
consequently provides a link between the analytic properties of the
zeta function and the distribution of primes in the integers. As the
simplest possible illustration of this link, we show how the
properties of the zeta function given above can be used to prove that
there are infinitely many primes.
If the set of primes in
were finite, then the Euler product
formula
A more sophisticated analysis of the zeta function along these lines can be used to prove both the analytic prime number theorem and Dirichlet's theorem on primes in arithmetic progressions 1. Proofs of the prime number theorem can be found in [2] and [5], and for proofs of Dirichlet's theorem on primes in arithmetic progressions the reader may look in [3] and [7].
A nontrivial zero of the Riemann zeta function is defined to be
a root
of the zeta function with the property that
. Any other zero is called trivial zero of
the zeta function.
The reason behind the terminology is as follows. For complex numbers
with real part greater than 1, the series definition (1)
immediately shows that no zeros of the zeta function exist in this
region. It is then an easy matter to use the functional
equation (3) to find all zeros of the zeta function
with real part less than 0 (it turns out they are exactly the values
, for
a positive integer). However, for values of
with
real part between 0 and 1, the situation is quite different, since we
have neither a series definition nor a functional equation to fall
back upon; and indeed to this day very little is known about the
behavior of the zeta function inside this critical strip of the
complex plane.
It is known that the prime number theorem is equivalent to the
assertion that the zeta function has no zeros with
or
. The celebrated Riemann hypothesis asserts that all nontrivial zeros
of the zeta function satisfy the much more precise equation
. If true, the hypothesis would have profound
consequences on the distribution of primes in the
integers [5].
This entry is a derivative of the Riemann zeta function article from PlanetMath. Author of the orginial article: djao. History page of the original is here
As of this snapshot date, this entry was owned by bloftin.