Editing Analytic number theory (section)

=== Riemann zeta function ===
{{Main|Riemann zeta function}}

[[Leonhard Euler|Euler]] showed that the [[fundamental theorem of arithmetic]] implies (at least formally) the ''[[Euler product]]''
: <math> \sum_{n=1}^\infty \frac {1}{n^s} = \prod_p^\infty \frac {1}{1-p^{-s}}\text{ for }s > 1</math>
where the product is taken over all prime numbers ''p''.

Euler's proof of the infinity of [[prime number]]s makes use of the divergence of the term at the left hand side for ''s'' = 1 (the so-called [[Harmonic series (mathematics)|harmonic series]]), a purely analytic result. Euler was also the first to use analytical arguments for the purpose of studying properties of integers, specifically by constructing [[generating function|generating power series]]. This was the beginning of analytic number theory.<ref name=":0">Iwaniec & Kowalski: Analytic Number Theory, AMS Colloquium Pub. Vol. 53, 2004</ref>

Later, Riemann considered this function for complex values of ''s'' and showed that this function can be extended to a [[meromorphic function]] on the entire plane with a simple [[Pole (complex analysis)|pole]] at ''s''&nbsp;=&nbsp;1.  This function is now known as the Riemann Zeta function and is denoted by ''ζ''(''s'').  There is a plethora of literature on this function and the function is a special case of the more general [[Dirichlet L-function]]s.

Analytic number theorists are often interested in the error of approximations such as the prime number theorem.  In this case, the error is smaller than ''x''/log&nbsp;''x''.  Riemann's formula for π(''x'') shows that the error term in this approximation can be expressed in terms of the zeros of the zeta function. In [[On the Number of Primes Less Than a Given Magnitude|his 1859 paper]], Riemann conjectured that all the "non-trivial" zeros of ζ lie on the line <math> \Re(s) = 1/2 </math> but never provided a proof of this statement.  This famous and long-standing conjecture is known as the ''[[Riemann Hypothesis]]'' and has many deep implications in number theory; in fact, many important theorems have been proved under the assumption that the hypothesis is true.  For example, under the assumption of the Riemann Hypothesis, the error term in the prime number theorem is {{nowrap|<math> O(x^{1/2+\varepsilon})</math>.}}

In the early 20th century [[G. H. Hardy]] and [[John Edensor Littlewood|Littlewood]] proved many results about the zeta function in an attempt to prove the Riemann Hypothesis.  In fact, in 1914,
Hardy proved that there were infinitely many zeros of the zeta function on the critical line
:<math> \Re(z) = 1/2. </math>

This led to several theorems describing the density of the zeros on the critical line.