Conjecture from Elliott and Halberstam

The conjecture of Elliott and Halberstam (EH, after Peter DTA Elliott and Heini Halberstam 1968) from analytic number theory concerns the error term in Dirichlet's theorem about the prime number distribution in arithmetic progressions .

Is the prime function (number of primes less than or equal ), and the number of prime numbers with (with relatively prime to ). According to Dirichlet's prime number theorem: ${\displaystyle \pi (x)}$ ${\displaystyle x}$ ${\displaystyle \pi (x,q,a)}$ ${\displaystyle y\leq x}$ ${\displaystyle y=a\mod q}$ ${\displaystyle a}$ ${\displaystyle q}$

${\displaystyle \pi (x;q,a)\approx {\frac {\pi (x)}{\varphi (q)}}}$

with Euler's phi function . Be ${\displaystyle \varphi }$

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

the error function of this distribution.

Elliott and Halberstam's conjecture is:

For each and there is a constant such that: ${\displaystyle \theta <1}$ ${\displaystyle A>0}$ ${\displaystyle C>0}$

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

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

For the conjecture is wrong, for it is part of the theorem of Bombieri and Vinogradow . ${\displaystyle \theta =1}$ ${\displaystyle \theta <{\frac {1}{2}}}$

As Dan Goldston , János Pintz and Cem Yıldırım showed, it follows from the conjecture that there are infinitely many pairs of prime numbers with a maximum distance of 16. James Maynard was also able to improve this to 12 assuming the guesswork. The Polymath project (Polymath 8, Terence Tao and others) was able to improve this to 6, assuming the generalized assumption of Elliott and Halberstam. Without using a conjecture, the best bound is currently (2019) 246 (see prime twin ).

Terry Tao showed in 2014 that the presumption of Vinogradov about the magnitude of the smallest quadratic non-residues (mod p) from the following Elliott Halberstam conjecture. Vinogradov's conjecture says for everyone . According to Linnik , Winogradov's conjecture also follows from the generalized Riemannian conjecture . ${\displaystyle n(p)}$ ${\displaystyle n(p)=O(p^{\varepsilon })}$ ${\displaystyle \epsilon >0}$

Generalized conjecture by Elliott and Halberstam

The generalized conjecture of Elliott and Halberstam (GEH) concerns the case that the Mangoldt function is not considered as in the theorem of Bombieri and Winogradow, but general arithmetic functions with certain additional properties, in particular they should be represented as Dirichlet convolution of two sequences (see below). Yōichi Motohashi (1976) showed that an analogue of the Bombieri and Winogradow theorem also applies here . Enrico Bombieri , John Friedlander and Henryk Iwaniec suspected that there is then also an analogue to the assumption of Elliott and Halberstam. The Elliott-Halberstam conjecture EH follows from GEH.

Be , two sequences of complex numbers and positive integers, so that for ${\displaystyle \alpha (m)}$ ${\displaystyle \beta (n)}$ ${\displaystyle M,N}$ ${\displaystyle x>2}$

• ${\displaystyle MN\leq x}$
• ${\displaystyle x^{\frac {1}{3}}<N\leq M<x^{\frac {2}{3}}}$
• The bearer of lies in and that of in ${\displaystyle \alpha (m)}$ ${\displaystyle (M,2M]}$ ${\displaystyle \beta (n)}$ ${\displaystyle (N,2N]}$
• ${\displaystyle \beta (n)}$ meets the Siegel-Walfisz condition:

${\displaystyle \left|\sum _{n\leq x \atop n=a\mod q}\beta (n)-{\frac {1}{\phi (q)}}\sum _{n\leq x \atop (n,q)=1}\beta (n)\right|=O({\frac {\|\beta \|{\sqrt {x}}}{{\log x}^{A}}})}$

for fixed and an integer with . is the complex Euclidean norm : ${\displaystyle A>0}$ ${\displaystyle a}$ ${\displaystyle (a,q)=1}$ ${\displaystyle \|\beta \|}$

${\displaystyle \|\beta \|={\sqrt {\sum _{n}{|\beta (n)|}^{2}}}}$

• For all natural numbers , and , where the number is a divisor of . ${\displaystyle n,m}$ ${\displaystyle |\alpha (m)|=O({d(m)}^{A}{\log(x)}^{B})}$ ${\displaystyle |\beta (n)|=O({d(n)}^{A}{\log(x)}^{B})}$ ${\displaystyle d(n)}$ ${\displaystyle n}$

The GEH is then the assumption that for a given , (with ) and for every integer : ${\displaystyle \epsilon >0}$ ${\displaystyle A>0}$ ${\displaystyle Q=x^{1-\epsilon }}$ ${\displaystyle a}$

${\displaystyle \sum _{q\leq Q \atop (q,a)=1}\left|\sum _{n=a\mod q}(\alpha *\beta )(n)-{\frac {1}{\phi (q)}}\sum _{(n,q)=1}(\alpha *\beta )(n)\right|=O({\frac {\|\alpha \|\|\beta \|{\sqrt {x}}}{{\log(x)}^{A}}})}$

Here is the Dirichlet convolution ${\displaystyle (\alpha *\beta )(n)=\sum _{ab=n}\alpha (a)\beta (b)}$

Assuming GEH, the Polymath project showed that at least one of the two sentences applies: The twin prime conjecture or the following “almost” Goldbach conjecture: Let n be a sufficiently large multiple of 6, then n or n + 2 is the sum of two prime numbers.

After the progress in the twin prime conjecture through the Polymath project, progress was also made on other problems under the assumption of GEH, for example by M. Ram Murty and Akshaa Vatwani in the Artin conjecture about primitive roots and the conjecture by Serge Lang and Hale Trotter for elliptic curves with complex multiplication.

Individual evidence

1. ^ Elliott, Halberstam, A conjecture in prime number theory, Symposia Mathematica IV, INDAM, Rome 1968, Academic Press 1970, pp. 59-72
2. ↑ John Friedlander, Andrew Granville, Limitations to the equidistribution of primes, Annals of Mathematics, Volume 129, 1989, pp. 363-382
3. ↑ Goldston, Pintz, Yildirim, Small gaps between primes exist
4. ↑ Maynard, Small gaps between primes, Annals of Mathematics, Volume 181, 2015, pp. 383-413
5. ↑ ^{a b} D.HJ Polymath: Variants of the Selberg sieve, and bounded intervals containing many primes, Research in the Mathematical Sciences, Volume 1, No. 12, 2014
6. ^ Tao, The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture, Algebra & Number Theory, Volume 9, 2015, pp. 1005-1034, Arxiv
7. ↑ Bombieri, Friedlander, Iwaniec, Primes in arithmetic progressions to large moduli, Acta Mathematica, Volume 156, 1986, pp. 203-251
8. ↑ Kevin Broughton, Equivalences of the Riemann Hypothesis, Cambridge UP 2017, p. 329
9. ↑ For a slightly different formulation see Polymath, Research in the Mathematical Sciences, Volume 1, 2014, No. 12
10. ↑ Ram Murty, Vatwani, A remark on the Lang-Trotter and Artin conjectures, Proc. American Mathematical Society, Volume 114, 2018, pp. 3191-3202