Elliott–Halberstam conjecture

{{Short description|On the distribution of prime numbers in arithmetic progressions}}

In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who stated a specific version of the conjecture in 1968.{{cite encyclopedia|year=1970|title=Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69)|publisher=Academic Press|location=London|pages=59–72|mr=0276195|first1=Peter D. T. A.|last1=Elliott|first2=Heini|last2=Halberstam|chapter=A conjecture in prime number theory}}

One version of the conjecture is as follows, and stating it requires some notation. Let $\backslash pi(x)$, the prime-counting function, denote the number of primes less than or equal to $x$. If $q$ is a positive integer and $a$ is coprime to $q$, we let $\backslash pi(x;q,a)$ denote the number of primes less than or equal to $x$ which are equal to $a$ modulo $q$. Dirichlet's theorem on primes in arithmetic progressions then tells us

that

:$\backslash pi(x;q,a)\; \backslash sim\; \backslash frac\{\backslash pi(x)\}\{\backslash varphi(q)\}\backslash \; \backslash \; (x\backslash rightarrow\backslash infty)$

where $\backslash varphi$ is Euler's totient function. If we then define the error function

:$E(x;q)\; =\; \backslash max\_\{\backslash text\{gcd\}(a,q)\; =\; 1\}\; \backslash left|\backslash pi(x;q,a)\; -\; \backslash frac\{\backslash pi(x)\}\{\backslash varphi(q)\}\backslash right|$

where the max is taken over all $a$ coprime to $q$, then the Elliott–Halberstam conjecture is the assertion that

for every and $A\; >\; 0$ there exists a constant $C\; >\; 0$ such that

:$\backslash sum\_\{1\; \backslash leq\; q\; \backslash leq\; x^\backslash theta\}\; E(x;q)\; \backslash leq\; \backslash frac\{C\; x\}\{\backslash log^A\; x\}$

for all $x\; >\; 2$.

This conjecture was proven for all by Enrico Bombieri{{cite journal |first=Enrico |last=Bombieri |title=On the large sieve |journal=Mathematika |volume=12 |year=1965 |issue=2 |pages=201–225 |doi= 10.1112/s0025579300005313 | mr=0197425}} and A. I. Vinogradov{{cite journal |first=Askold Ivanovich |last=Vinogradov |title=The density hypothesis for Dirichlet L-series |language=ru |journal=Izv. Akad. Nauk SSSR Ser. Mat. |volume=29 |issue=4 |year=1965 |pages=903–934 |mr=197414 }} Corrigendum. ibid. 30 (1966), pages 719-720. (Russian) (the Bombieri–Vinogradov theorem, sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the generalized Riemann hypothesis. It is known that the conjecture fails at the endpoint $\backslash theta\; =\; 1$.{{cite journal | last1=Friedlander | first1=John | last2=Granville | first2=Andrew | title=Limitations to the equi-distribution of primes I | journal=Annals of Mathematics | volume=129 | issue=2 | year=1989 | pages=363–382 | mr=0986796 | doi=10.2307/1971450| jstor=1971450 }} In 1986, Bombieri, Friedlander and Iwaniec generalized the Elliott-Halberstam conjecture, using Dirichlet convolution of arithmetic functions related to the von Mangoldt function.{{cite journal | last=Bombieri | first=E. | last2=Friedlander | first2=J. B. | last3=Iwaniec | first3=H. | title=Primes in arithmetic progressions to large moduli | journal=Acta Mathematica | publisher=International Press of Boston | volume=156 | issue=1 | year=1986 | issn=0001-5962 | doi=10.1007/bf02399204 | doi-access=free | pages=203–251}}

The Elliott–Halberstam conjecture has several consequences. A striking one is the result announced by Dan Goldston, János Pintz, and Cem Yıldırım,{{cite journal|arxiv=math.NT/0508185|last1=Goldston|first1=D. A.|last2=Pintz|first2=J.|last3=Yıldırım|first3=C. Y.|title=Primes in Tuples I|journal=Annals of Mathematics|series=Second Series|pages=819–862|volume=170|date=2009|issue=2|doi=10.4007/annals.2009.170.819|doi-access=free}}
{{cite journal|arxiv=math.NT/0505300|last1=Goldston|first1=D. A.|last2=Motohashi|first2=Y.|last3=Pintz|first3=J.|last4=Yıldırım|first4=C. Y.|title=Small Gaps between Primes Exist|journal=Proceedings of the Japan Academy, Series A, Mathematical Sciences|volume=82|issue=4|pages=61–65|date=April 2006|doi=10.3792/pjaa.82.61|doi-access=free}}
{{cite journal|arxiv=math.NT/0506067|last1=Goldston|first1=D. A.|last2=Graham|first2=S. W.|last3=Pintz|first3=J.|last4=Yıldırım|first4=C. Y.|title=Small gaps between primes or almost primes|journal=Transactions of the American Mathematical Society|volume=361|issue=10|date=2009|pages=5285–5330|doi=10.1090/S0002-9947-09-04788-6|doi-access=free}}{{cite journal |first=Kannan |last=Soundararajan |author-link=Kannan Soundararajan |title=Small gaps between prime numbers: The work of Goldston–Pintz–Yıldırım |journal=Bulletin of the American Mathematical Society |volume=44 |year=2007 |issue=1 |pages=1–18 |doi=10.1090/S0273-0979-06-01142-6 | mr=2265008|arxiv=math/0605696 |s2cid=119611838 }} which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16. In November 2013, James Maynard showed that subject to the Elliott–Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 12.{{cite journal | title=Small gaps between primes | last=Maynard | first=James | author-link=James Maynard (mathematician) | year=2015 | journal=Annals of Mathematics | volume=181 | issue=1 | pages=383–413 | mr=3272929 | doi=10.4007/annals.2015.181.1.7| arxiv=1311.4600 | s2cid=55175056 }} In August 2014, Polymath group showed that subject to the generalized Elliott–Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6.{{cite journal | author=D.H.J. Polymath | title=Variants of the Selberg sieve, and bounded intervals containing many primes | journal=Research in the Mathematical Sciences | volume=1 | number=12 | doi=10.1186/s40687-014-0012-7 | arxiv=1407.4897 | year=2014 | mr=3373710| s2cid=119699189 | doi-access=free }} Without assuming any form of the conjecture, the lowest proven bound is 246.