Elliott–Halberstam conjecture

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.[1]

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

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

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

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

where the max is taken over all ${\displaystyle a}$ coprime to ${\displaystyle q}$, then the Elliott–Halberstam conjecture is the assertion that for every ${\displaystyle \theta <1}$ and ${\displaystyle A>0}$ there exists a constant ${\displaystyle C>0}$ such that

${\displaystyle \sum _{1\leq q\leq x^{\theta }}E(x;q)\leq {\frac {Cx}{\log ^{A}x}}}$

for all ${\displaystyle x>2}$.

This conjecture was proven for all ${\displaystyle \theta <1/2}$ by Enrico Bombieri[2] and A. I. Vinogradov[3] (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 ${\displaystyle \theta =1}$.[4] In 1986, Bombieri, Friedlander and Iwaniec generalized the Elliott-Halberstam conjecture, using Dirichlet convolution of arithmetic functions related to the von Mangoldt function.[5]

The Elliott–Halberstam conjecture has several consequences. One striking one is the result announced by Dan Goldston, János Pintz, and Cem Yıldırım,[6][7] 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.[8] 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.[9] Without assuming any form of the conjecture, the lowest proven bound is 246.

Original conjecture

The original Elliott-Halberstam conjecture is not clearly stated in their paper,[1] but can be inferred there from (1) page 59 and the comment above the Theorem on page 62. It says that

${\displaystyle \sum _{q\leq x}\varphi (q)\max _{(h,q)=1}\left(\pi (x,q,h)-{\frac {\operatorname {li} x}{\varphi (q)}}\right)^{2}\ll {\frac {x^{2}}{\log ^{A}x}}}$

provided ${\displaystyle X<x(\log x)^{-A-1}}$, where ${\displaystyle \operatorname {li} x}$ denotes the logarithmic integral and ${\displaystyle \varphi }$ the Euler function.

See also

• Barban–Davenport–Halberstam theorem
• Sexy prime
• Siegel–Walfisz theorem

Notes

1. Elliott, Peter D. T. A.; Halberstam, Heini (1970). "A conjecture in prime number theory". Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69). London: Academic Press. pp. 59–72. MR 0276195.
2. Bombieri, Enrico (1965). "On the large sieve". Mathematika. 12 (2): 201–225. doi:10.1112/s0025579300005313. MR 0197425.
3. Vinogradov, Askold Ivanovich (1965). "The density hypothesis for Dirichlet L-series". Izv. Akad. Nauk SSSR Ser. Mat. (in Russian). 29 (4): 903–934. MR 0197414. Corrigendum. ibid. 30 (1966), pages 719-720. (Russian)
4. Friedlander, John; Granville, Andrew (1989). "Limitations to the equi-distribution of primes I". Annals of Mathematics. 129 (2): 363–382. doi:10.2307/1971450. JSTOR 1971450. MR 0986796.
5. Bombieri, E.; Friedlander, J. B.; Iwaniec, H. (1986). "Primes in arithmetic progressions to large moduli". Acta Mathematica. 156 (1). International Press of Boston: 203–251. doi:10.1007/bf02399204. ISSN 0001-5962.
6. Goldston, D. A.; Pintz, J.; Yıldırım, C. Y. (2009). "Primes in Tuples I". Annals of Mathematics. Second Series. 170 (2): 819–862. arXiv:math.NT/0508185. doi:10.4007/annals.2009.170.819.
   Goldston, D. A.; Motohashi, Y.; Pintz, J.; Yıldırım, C. Y. (April 2006). "Small Gaps between Primes Exist". Proceedings of the Japan Academy, Series A, Mathematical Sciences. 82 (4): 61–65. arXiv:math.NT/0505300. doi:10.3792/pjaa.82.61.
   Goldston, D. A.; Graham, S. W.; Pintz, J.; Yıldırım, C. Y. (2009). "Small gaps between primes or almost primes". Transactions of the American Mathematical Society. 361 (10): 5285–5330. arXiv:math.NT/0506067. doi:10.1090/S0002-9947-09-04788-6.
7. Soundararajan, Kannan (2007). "Small gaps between prime numbers: The work of Goldston–Pintz–Yıldırım". Bulletin of the American Mathematical Society. 44 (1): 1–18. arXiv:math/0605696. doi:10.1090/S0273-0979-06-01142-6. MR 2265008. S2CID 119611838.
8. Maynard, James (2015). "Small gaps between primes". Annals of Mathematics. 181 (1): 383–413. arXiv:1311.4600. doi:10.4007/annals.2015.181.1.7. MR 3272929. S2CID 55175056.
9. D.H.J. Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences. 1 (12). arXiv:1407.4897. doi:10.1186/s40687-014-0012-7. MR 3373710. S2CID 119699189.