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Patterns of Primes in Arithmetic Progressions PATTERNS OF PRIMES IN ARITHMETIC PROGRESSIONS JÁNOS PINTZ Rényi Institute of the Hungarian Academy of Sciences CIRM, Dec. 13, 2016 2 1. Patterns of primes th 1 Notation: pn the n prime, P = fpi gi=1, dn = pn+1 − pn. Abbreviation: i.o. = infinitely often, Z+ = f1; 2;::: g 2 Twin Prime Conjecture fn; n + 2g 2 P i.o. () dν = 2 i.o. Polignac Conjecture (1849) 2 j h −! dn = h i.o. Definition k H = Hk = fhi gi=1, 0 ≤ h1 < h2 < hk is admissible if the number of residue classes covered by H mod p, νp(H) < p for every prime p. Dickson Conjecture (1904). Hk admissible k k =) fn + hi gi=1 2 P i.o. 2 / 35 3 Hardy–Littlewood Conjecture (1923). H = Hk admissible =) X x 1 ∼ k σ(H); n≤x log x k fn+hi g2P −k Y νp(H) 1 σ(H) = 1 − 1 − p p p 3 / 35 4 Remark 1. HL conjecture implies: Strong HL conjecture: H = Hk admissible =) X x 1 ∼ σ(H) (log x)k n≤x k fn+hi g2P n+h2P= if h2[0;hk ]; h6=hi Proof. By Selberg’s upper bound sieve X x 1 : (log x)k+1 n≤x k fn+hi g2P n+h2P Remark 2. Dickson Conjecture 6=) Strong Dickson Conjecture. 4 / 35 5 2. Primes in Arithmetic Progressions (AP) Conjecture (Lagrange, Waring, Erdős–Turán (1936)). The primes contain k-term AP’s for every k. Conjecture (Erdős–Turán (1936)). If A ⊂ Z+ has positive upper density then A contains k-term AP’s for every k. Roth (1953): This is true for k = 3. Szemerédi (1975) This is true for every k. 5 / 35 6 3. History before 2000 (2004) Erdős (1940) dn < (1 − c0) log n i.o. c0 > 0 fix. Bombieri–Davenport (1966) dn < (log n)=2 i.o. H. Maier (1988) dn < (log n)=4 i.o. Van der Corput (1939) P contains infinitely many 3-term AP’s. Heath-Brown (1984) There are infinitely many pairs n; d such that n; n + d; n + 2d 2 P, n + 3d 2 P2. Definition n = Pk -number if it has at most k prime factors (ALMOST PRIMES). 6 / 35 7 4. Results after 2000 Green–Tao Theorem (2004–2008): P contains k-term AP’s for every k. Goldston–Pintz–Yıldırım (2005–2009): lim inf dn= log n = 0. n!1 c 1 GPY (2006–2010): lim inf dn=(log n) = 0 if c > . n!1 2 6 Zhang (2013–2014): Hk admissible, k > 3:5 · 10 =) n + Hk contains at least 2 primes i.o. 7 / 35 8 Maynard (2013–2015): This is true for k ≥ 105. Polymath (2014): This is true for k ≥ 50. Maynard (2013–2015), Tao unpublished: Hk admissible 1 =) n + Hk contains at least 4 + o(1) log k primes i.o. 8 / 35 9 5. Patterns of primes in arithmetic progressions A common generalization of the Green–Tao and Maynard–Tao theorem is Theorem 1 (J. P. 2017) Let m > 0 and A = fa1;:::; ar g be a set of r distinct integers with r sufficiently large depending on m. Let N(A) denote the number of integer m-tuples fh1;:::; hmg ⊆ A such that there exist for every ` infinitely many `-term arithmetic progressions ` th of integers fni gi=1 where ni + hj is the j prime following ni prime for each pair i; j. Then m m (5.1) N(A) m # (h1;:::; hm) 2 A m jAj = r : 9 / 35 10 Theorem 1 will follow by the application of Maynard’s method from the weaker Theorem 2 (J. P. 2017) Let m be a positive integer, H = fh1;:::; hk g be an admissible set of k distinct non-negative integers hi 6 H, k = dCm2e4me with a sufficiently large absolute constant C. Then there exists an m-element subset 0 0 0 (5.2) fh1; h2;:::; hmg ⊆ H such that for every positive integer ` we have infinitely many `-element non-trivial arithmetic progressions of integers ni 0 such that ni + hj 2 P for 1 6 i 6 `, 1 6 j 6 m, further 0 th ni + hj is always the j prime following ni . 10 / 35 11 In fact we prove a stronger result, namely Theorem 3 (J. P. 2017) 2 4k0 There is some C, such that for all k0 and all k > Ck0 e there is some c > 0, such that for all admissible tuples fh1;:::; hk g the number N(x) of integers n ≤ x, such that c n + hi is n -pseudo prime, and among these k integers there are at least k primes, satisfies N(x) x . These N(x) 0 logk x integers n ≤ x contain an m-term AP if x > C0(m). 11 / 35 12 6. Structure of the proof of the Maynard–Tao theorem (i) Key parameters: H = fh1;:::; hk g given (0 ≤ h1 < h2 < : : : < hk ) N large, we look for primes of the form n + hi with n 2 [N; 2N) R = Nθ=2−", where θ is a level of distribution of primes: X π(x) x (6.1) max π(x; q; a) − A a '(q) (log x)A θ (a;q)=1 q6x holds for any A > 0 where the symbol of Vinogradov means that f (x) = O(g(x)) is abbreviated by f (x) g(x). Remark. θ = 1=2 admissible: Bombieri–Vinogradov theorem (1965). θ > 0 Rényi (1947) Y ∗ (6.2) W = p D0 = C (k) suitably large p≤D0 (6.3) n ≡ ν0(mod W )(ν0 + hi ; W ) = 1 for i = 1;:::; k: 12 / 35 13 (ii) We weight the numbers n ≡ ν0 (mod W ), n 2 [N; 2N) by wn, so that wn ≥ 0 and on average wn would be large if we k have many primes among fn + hi gi=1, 2 X (6.4) wn = λd1;:::;dk : di jn+hi 8i k Q 2 k µ ri Y X i=1 (6.5) λ = µ(d )d y d1;:::;dk i i k r1;:::;rk i=1 r1;:::;rk Q '(ri ) di jri 8i i=1 (ri ;W )=1 k Q whenever di ; W = 1 and λd1;:::;dr = 0 otherwise. i=1 13 / 35 14 log r log r (6.6) y = F 1 ;:::; k r1;:::;rk log R log R where F is piecewise differentiable, real, F and F 0 bounded, supported on ( k ) k X (6.7) Rk = (x1;:::; xk ) 2 [0; 1] : xi 6 1 : i=1 (iii) Let χP (n) denote the characteristic function of P, (6.8) k X X X S1 := wn; S2 := wn χP (n+hi ) ; n n i=1 N6n<2N N6n<2N n≡ν0 (mod W ) n≡ν0 (mod W ) 14 / 35 15 If we succeed to choose F , thereby λd and wn in such a way + that rk 2 Z (6.9) S2 > S1; or S2 > (rk − 1)S1 resp. we obtain at lest two, or k0 primes, resp. among n + h1;:::; n + hk =) bounded gaps between primes or even k0 primes in bounded intervals i.o. 15 / 35 16 (iv) First step towards this: evaluation of S1 and S2. Proposition 1. We have as N ! 1 1 k k 1 + O D '(W ) N(log R) (6.10) S = 0 I (F ); 1 W k+1 k 1 + O 1 '(W )k N(log R)k+1 k D0 X (j) (6.11) S = J (F ); 2 W k+1 k j=1 1 1 Z Z 2 (6.12) Ik (F ) = ::: F (t1;:::; tk ) dt1 ::: dtk ; 0 0 (6.13) 1 1 1 Z Z Z 2 (j) Jk (F ) = ::: F (t1;:::; tk )dtj dt1 ::: dtj−1dtj+1 ::: dtk : 0 0 0 16 / 35 17 After this we immediately obtain Corollary. If the sup is taken with Fk as before and k P J(j)(F ) k j=1 θMk (6.14) Mk = sup ; rk = Ik (F ) 2 and let H be a fixed admissible sequence H = fh1;:::; hk g of size k. Then there are infinitely many integers n such that at least rk of the n + hi (1 6 i 6 k) are simultaneously primes. Proposition 2. M105 > 4 and Mk > log k − 2 log log k − 2 for k > k0. 17 / 35 18 7. A stronger version of the Maynard–Tao theorem Theorem 3 gives a stronger form in 3 aspects: (i) all numbers n + hi are almost primes, having all prime factors greater than nc(k); c0(k)N (ii) the number of such n’s is at least , the true order logk N of magnitude of n 2 [N; 2N) with all n + hi being nc -almost primes; (iii) for every ` we have (infinitely) many `-term AP’s with the same prime pattern. Remark: properties (i) and (ii) are interesting in themselves, but crucial in many applications, in particular in showing (iii). 18 / 35 19 Let P−(n) denote the smallest prime factor of n. The following Lemma shows that the contribution of n’s to S1 Q c1(k) with at least one prime p j (n + hi ), p < n is negligible if θ=2−" c1(k) is suitably small R = N . Lemma 1. We have X c1(k) log N (7.1) S − = w S : 1 n k;H log R 1 N6n<2N n≡ν0 (mod W ) k − Q c (k) P (n+hi ) <n 1 i=1 19 / 35 20 Corollary. We get immediately property (i). Further 2 2 2k 2k (7.2) wn λmax ymax(log R) (log R) Q since (n + hi ) has in this case just a bounded number of prime factors, so the sum over the divisors can be substituted by the largest term (apart from a factor depending on k).
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