Editing Number theory (section)

=== Analytic number theory ===
{{Main|Analytic number theory}}

[[File:Complex zeta.jpg|thumb|[[Riemann zeta function]] ζ(''s'') in the [[complex plane]]. The color of a point ''s'' gives the value of ζ(''s''): dark colors denote values close to zero and hue gives the value's [[Argument (complex analysis)|argument]].]]
[[File:ModularGroup-FundamentalDomain.svg|thumb|The action of the [[modular group]] on the [[upper half plane]]. The region in grey is the standard [[fundamental domain]].]]

Analytic number theory may be defined
* in terms of its tools, as the study of the integers by means of tools from [[Real analysis|real]] and [[Complex analysis|complex]] analysis;{{sfn|Apostol|1976|p=7}} or
* in terms of its concerns, as the study within number theory of estimates on the size and density of certain numbers (e.g., primes), as opposed to identities.<ref>{{harvnb|Granville|2008|loc=section 1}}: "The main difference is that in algebraic number theory [...] one typically considers questions with answers that are given by exact formulas, whereas in analytic number theory [...] one looks for ''good approximations''."</ref>

Some subjects generally considered to be part of analytic number theory (e.g., [[sieve theory]]) are better covered by the second rather than the first definition.<ref group="note">Sieve theory figures as one of the main subareas of analytic number theory in many standard treatments; see, for instance, {{harvnb|Iwaniec|Kowalski|2004}} or {{harvnb|Montgomery|Vaughan|2007}}</ref> Small sieves, for instance, use little analysis and yet still belong to analytic number theory.<ref group="note">This is the case for some combinatorial sieves such as the [[Brun sieve]], rather than for [[Large sieve|large sieves]]. The study of the latter now includes ideas from [[Harmonic analysis|harmonic]] and [[functional analysis]].</ref>

The following are examples of problems in analytic number theory: the [[prime number theorem]], the [[Goldbach conjecture]], the [[twin prime conjecture]], the [[Hardy–Littlewood conjecture]]s, the [[Waring problem]] and the [[Riemann hypothesis]]. Some of the most important tools of analytic number theory are the [[circle method]], [[sieve theory|sieve methods]] and [[L-functions]] (or, rather, the study of their properties). The theory of [[modular form]]s (and, more generally, [[automorphic forms]]) also occupies an increasingly central place in the toolbox of analytic number theory.<ref>See the remarks in the introduction to {{harvnb|Iwaniec|Kowalski|2004|p=1}}: "However much stronger...".</ref>

One may ask analytic questions about [[algebraic number]]s, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define [[prime ideal]]s (generalizations of [[prime number]]s in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question [[Landau prime ideal theorem|can be answered]] by means of an examination of [[Dedekind zeta function]]s, which are generalizations of the [[Riemann zeta function]], a key analytic object at the roots of the subject.<ref>{{harvnb|Granville|2008|loc=section 3}}: "[Riemann] defined what we now call the Riemann zeta function [...] Riemann's deep work gave birth to our subject [...]"</ref> This is an example of a general procedure in analytic number theory: deriving information about the distribution of a [[sequence]] (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function.<ref name=":0">See, for example, {{harvnb|Montgomery|Vaughan|2007}}, p. 1.</ref>