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, in contrast to elementary number theory, relies on [[complex numbers]] and techniques from analysis and calculus. 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>
It studies the distribution of primes, behavior of number-theoric functions, and irrational numbers.<ref>{{Cite web |last=Karatsuba |first=A.A. |date=2014-10-18 |title=Analytic number theory |url=https://encyclopediaofmath.org/wiki/Analytic_number_theory |website=Encyclopedia of Mathematics}}</ref>

Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, many of the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics.<ref>See, for example, the initial comment in {{harvnb|Iwaniec|Kowalski|2004|p=1}}.</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>

[[Mathematical analysis|Analysis]] is the branch of mathematics that studies the [[Limit (mathematics)|limit]], defined as the value to which a sequence or function tends as the argument (or index) approaches a specific value. For example, the limit of the sequence 0.9, 0.99, 0.999, ... is 1. In the context of functions, the limit of <math display="inline">\frac1x</math> as <math>x</math> approaches infinity is 0.<ref>{{Cite book |last=Tanton |first=James |title=Encyclopedia of Mathematics |chapter=Limit}}</ref> The complex numbers extend the real numbers with the imaginary unit <math>i</math> defined as the solution to <math>i^2 = -1</math>. Every complex number can be expressed as <math>x + iy</math>, where <math>x</math> is called the real part and <math>y</math> is called the imaginary part.<ref>{{Cite book |last=Weisstein |first=Eric W. |title=CRC Concise Encyclopedia of Mathematics |year=2002 |chapter=Complex Numbers}}</ref>

The [[distribution of primes]], described by the function <math>\pi</math> that counts all primes up to a given real number, is unpredictable and is a major subject of study in number theory. Elementary formulas for a partial sequence of primes, including [[Lucky numbers of Euler|Euler's prime-generating polynomials]] have been developed. However, these cease to function as the primes become too large. The prime number theorem in analytic number theory provides a formalisation of the notion that prime numbers appear less commonly as their numerical value increases. One distribution states, informally, that the function <math>\frac{x}{\log(x)}</math> approximates <math>\pi(x)</math>. Another distribution involves an offset logarithmic integral which converges to <math>\pi(x)</math> more quickly.<ref name=":5" />
[[File:Riemann_Explicit_Formula.gif|thumb|Corrections to an [[Prime-counting function#Exact form|estimate]] of the prime-counting function using zeros of the zeta function.]]
The [[zeta function]] has been demonstrated to be connected to the distribution of primes. It is defined as the series<math display="block"> \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots</math>that converges if <math> s</math> is greater than 1. Euler demonstrated a link involving the infinite product over all prime numbers, expressed as the identity <math display="block">\zeta(s) = \prod_{p \text{ prime}} \left(1 - \frac{1}{p^{s}}\right)^{-1}.</math>Riemann extended the definition to a complex variable and conjectured that all nontrivial cases (<math>0 < \Re(s) < 1</math>) where the function returns a zero are those in which the real part of <math>s</math> is equal to <math display="inline">\frac12</math>. He established a connection between the nontrivial zeroes and the prime-counting function. In what is now recognised as the unsolved [[Riemann hypothesis]], a solution to it would imply direct consequences for understanding the distribution of primes.<ref>{{Cite book |last=Tanton |first=James |title=Encyclopedia of Mathematics |year=2005 |chapter=Zeta function}}</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>

Elementary number theory works with ''[[elementary proof|elementary proofs]]'', a term that excludes the use of [[complex numbers]] but may include basic analysis.<ref name=":2" /> For example, the [[prime number theorem]] was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by [[Paul Erdős|Erdős]] and [[Atle Selberg|Selberg]].{{sfn|Goldfeld|2003}} The term is somewhat ambiguous. For example, proofs based on complex [[Tauberian theorem]]s, such as [[Wiener–Ikehara theorem|Wiener–Ikehara]], are often seen as quite enlightening but not elementary despite using [[Fourier analysis]], not complex analysis. Here as elsewhere, an ''elementary'' proof may be longer and more difficult for most readers than a more advanced proof.

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>