Editing Analytic number theory (section)

=== Multiplicative number theory ===
{{main|Multiplicative number theory}}
[[Euclid]] showed that there are infinitely many prime numbers. An important question is to determine the asymptotic distribution of the prime numbers; that is, a rough description of how many primes are smaller than a given number. [[Carl Gauss|Gauss]], amongst others, after computing a large list of primes, conjectured that the number of primes less than or equal to a large number ''N'' is close to the value of the [[integral]]

<math display="block">\int^N_2 \frac{1}{\log t} \, dt.</math>

In 1859 [[Bernhard Riemann]] used complex analysis and a special [[meromorphic]] function now known as the [[Riemann zeta function]] to derive an analytic expression for the number of primes less than or equal to a real number&nbsp;''x''.  Remarkably, the main term in Riemann's formula was exactly the above integral, lending substantial weight to Gauss's conjecture.  Riemann found that the error terms in this expression, and hence the manner in which the primes are distributed, are closely related to the complex zeros of the zeta function. Using Riemann's ideas and by getting more information on the zeros of the zeta function, [[Jacques Hadamard]] and [[Charles Jean de la Vallée-Poussin]] managed to complete the proof of Gauss's conjecture. In particular, they proved that if 
<math display="block">\pi(x) = (\text{number of primes }\leq x),</math>
then
<math display="block">\lim_{x \to \infty} \frac{\pi(x)}{x/\log x} = 1.</math>

This remarkable result is what is now known as the ''[[prime number theorem]]''. It is a central result in analytic number theory. Loosely speaking, it states that given a large number ''N'', the number of primes less than or equal to ''N'' is about ''N''/log(''N'').

More generally, the same question can be asked about the number of primes in any [[arithmetic progression]] ''a'' + ''nq'' for any integer ''n''. In one of the first applications of analytic techniques to number theory, Dirichlet proved that any arithmetic progression with ''a'' and ''q'' coprime contains infinitely many primes. The prime number theorem can be generalised to this problem; letting 
<math display="block">\pi(x, a, q) = (\text {number of primes } \leq x \text{ in the arithmetic progression } a + nq, \ n \in \mathbf Z), </math>
then if ''a'' and ''q'' are coprime,
<math display="block">\lim_{x \to \infty} \frac{\pi(x,a,q)\phi(q)}{x/\log x} = 1,</math>
where <math>\phi</math> is the [[totient function]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Totient Function |url=https://mathworld.wolfram.com/TotientFunction.html |access-date=2025-02-09 |website=mathworld.wolfram.com |language=en}}</ref>

There are also many deep and wide-ranging conjectures in number theory whose proofs seem too difficult for current techniques, such as the [[Twin prime|twin prime conjecture]] which asks whether there are infinitely many primes ''p'' such that ''p''&nbsp;+&nbsp;2 is prime. On the assumption of the [[Elliott–Halberstam conjecture]] it has been proven recently that there are infinitely many primes ''p'' such that ''p''&nbsp;+&nbsp;''k'' is prime for some positive even ''k'' at most&nbsp;12. Also, it has been proven unconditionally (i.e. not depending on unproven conjectures) that there are infinitely many primes ''p'' such that ''p''&nbsp;+&nbsp;''k'' is prime for some positive even ''k'' at most&nbsp;246.