Editing Prime number theorem (section)

== Elementary proofs ==
In the first half of the twentieth century, some mathematicians (notably [[G. H. Hardy]]) believed that there exists a hierarchy of proof methods in mathematics depending on what sorts of numbers ([[integer]]s, [[real number|reals]], [[complex number|complex]]) a proof requires, and that the prime number theorem (PNT) is a "deep" theorem by virtue of requiring [[complex analysis]].<ref name="Goldfeld Historical Perspective">{{cite book | first=Dorian | last=Goldfeld | chapter-url=http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf | chapter=The elementary proof of the prime number theorem: an historical perspective | year=2004 | title=Number theory (New York, 2003) | pages=179–192 | mr=2044518 | editor1-last=Chudnovsky | editor1-first=David | editor2-last=Chudnovsky | editor2-first=Gregory | editor3-last=Nathanson | editor3-first=Melvyn | location=New York | publisher=Springer-Verlag | isbn=978-0-387-40655-8  | doi=10.1007/978-1-4419-9060-0_10}}</ref> This belief was somewhat shaken by a proof of the PNT based on [[Wiener's tauberian theorem]], though Wiener's proof ultimately relies on properties of the Riemann zeta function on the line <math>\text{re}(s)=1</math>, where complex analysis must be used.

In March 1948, [[Atle Selberg]] established, by "elementary" means, the [[Selberg's identity|asymptotic formula]]
: <math>\vartheta ( x )\log ( x ) + \sum\limits_{p \le x} {\log ( p )}\ \vartheta \left( {\frac{x}{p}} \right) = 2x\log ( x ) + O( x )</math>
where
: <math>\vartheta ( x ) = \sum\limits_{p \le x} {\log ( p )}</math>
for primes {{mvar|p}}.<ref name="Selberg1949">{{citation|last=Selberg|first=Atle|title=An Elementary Proof of the Prime-Number Theorem|journal=[[Annals of Mathematics]]|year=1949|volume=50|issue=2|pages=305–313|doi=10.2307/1969455|mr=0029410|jstor=1969455|s2cid=124153092 }}</ref> By July of that year, Selberg and [[Paul Erdős]]<ref name="Erdős1949" /> had each obtained elementary proofs of the PNT, both using Selberg's asymptotic formula as a starting point.<ref name="Goldfeld Historical Perspective"/><ref name=interview>{{cite journal|url=https://www.ams.org/bull/2008-45-04/S0273-0979-08-01223-8/S0273-0979-08-01223-8.pdf |first1=Nils A.|last1= Baas|first2= Christian F.|last2= Skau |journal= Bull. Amer. Math. Soc. |volume=45 |year=2008|pages= 617–649 |title=The lord of the numbers, Atle Selberg. On his life and mathematics|doi=10.1090/S0273-0979-08-01223-8|issue=4|mr=2434348|doi-access=free}}</ref> These proofs effectively laid to rest the notion that the PNT was "deep" in that sense, and showed that technically "elementary" methods were more powerful than had been believed to be the case. On the history of the elementary proofs of the PNT, including the Erdős–Selberg [[priority dispute]], see an article by [[Dorian Goldfeld]].<ref name="Goldfeld Historical Perspective" />

There is some debate about the significance of Erdős and Selberg's result. There is no rigorous and widely accepted definition of the notion of [[elementary proof]] in number theory, so it is not clear exactly in what sense their proof is "elementary". Although it does not use complex analysis, it is in fact much more technical than the standard proof of PNT. One possible definition of an "elementary" proof is "one that can be carried out in first-order [[Peano arithmetic]]." There are number-theoretic statements (for example, the [[Paris–Harrington theorem]]) provable using [[second order arithmetic|second order]] but not [[first-order arithmetic|first-order]] methods, but such theorems are rare to date. Erdős and Selberg's proof can certainly be formalized in Peano arithmetic, and in 1994, Charalambos Cornaros and Costas Dimitracopoulos proved that their proof can be formalized in a very weak fragment of PA, namely {{math|''I''Δ<sub>0</sub> + exp}}.<ref>{{cite journal|last1=Cornaros|first1=Charalambos|last2=Dimitracopoulos|first2=Costas|title=The prime number theorem and fragments of ''PA''|year=1994|url=http://mpla.math.uoa.gr/~cdimitr/files/publications/AML_33.pdf|journal=Archive for Mathematical Logic|volume=33|issue=4|pages=265–281|doi=10.1007/BF01270626|mr=1294272|s2cid=29171246|url-status=dead|archive-url=https://web.archive.org/web/20110721083756/http://mpla.math.uoa.gr/~cdimitr/files/publications/AML_33.pdf|archive-date=2011-07-21}}</ref> However, this does not address the question of whether or not the standard proof of PNT can be formalized in PA.

A more recent "elementary" proof of the prime number theorem uses [[ergodic theory]], due to Florian Richter.<ref>Bergelson, V., & Richter, F. K. (2022). Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions. Duke Mathematical Journal, 171(15), 3133-3200.</ref>  The prime number theorem is obtained there in an equivalent form that the [[Cesàro sum]] of the values of the [[Liouville function]] is zero.  The Liouville function is <math>(-1)^{\omega(n)}</math> where <math>\omega(n)</math> is the number of prime factors, with multiplicity, of the integer <math>n</math>.  Bergelson and Richter (2022) then obtain this form of the prime number theorem from an [[ergodic theorem]] which they prove:
: Let <math>X</math> be a compact metric space, <math>T</math> a continuous self-map of <math>X</math>, and <math>\mu</math> a <math>T</math>-invariant Borel probability measure for which <math>T</math> is [[uniquely ergodic]].  Then, for every <math>f\in C(X)</math>,
<math display="block">\tfrac1N\sum_{n=1}^Nf(T^{\omega(n)}x)\to \int_Xf\,d\mu,\quad\forall x\in X.</math>

This ergodic theorem can also be used to give "soft" proofs of results related to the prime number theorem, such as the  [[Pillai–Selberg theorem]] and [[Erdős–Delange theorem]].