Analytic number theory
Analytic number theory [ edit ] Main article: Analytic number theory 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 . 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 and complex analysis; [70] or in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities. [79] Some subjects generally considered to be part of analytic number theory, for example, sieve theory , [note 10] are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis, [note 11] ...