The Riemann Zeta Function
In the late 17th century it was known that the harmonic series
is divergent, as it was proved by Pietro Mengoli, Johann Bernoulli and Jakob Bernoulli. Euler considered the series of the form:
for integer values of , which can be easily shown to converge
Posteriorly Dirichlet and Chebyshev would consider real valued exponents. It is also easy to prove that is a convergent series. Riemann went one step further and considered a complex variable and proved that converges for and it can be continued analitically to all complex values . This extension of
is well defined and holomorphic in , it is denoted and is called the Riemann Zeta Function.
Some properties of
The zeta function can be written for as the sum of a series and the Euler product formula allows us to write it as an infinite product:
One of the steps in proving that is holomorphic in consists in verifiying the following functional equation
where is the gamma function . The fact that it verifies this functional equation allows to prove many other interesting properties.
In his original paper, Riemann suggested that all zeroes of which are not trivial ( ) are in the line
This came to be known as the Riemann Hypothesis and it's been an open problem for over 150 years.
Proving the Riemann Hypothesis would have many significant consequences. One of the most important ones is the precise caracterization of the distribution of prime numbers. Let's take the prime counting function , which gives for each the number of prime numbers less than or equal to . The Prime Number Theorem states that
This theorem was originally proved independently by Hadamard and de la Valle Poussin. If we consider
then the Prime Number Theorem can be written as
Von Koch proved that if and only if the Riemann hypothesis is true then
and Schoenfeld stated more precisely that the Riemann hypothesis is equivalent to
for all and this is indeed the tightest limit possible.