Probabilistic models for primes and large gaps William Banks Kevin Ford Terence Tao July, 2019 Banks, Ford, Tao Probabilistic models for primes 1 / 28 July, 2019 1 / 28 Large gaps between primes th Def: G(x) = max(pn − pn−1), pn is the n prime. pn6x 2; 3; 5; 7;:::; 109; 113; 127; 131;:::; 9547; 9551; 9587; 9601;::: Upper bound: G(x) = O(x0:525) (Baker-Harman-Pintz, 2001). Improve to O(x1/2 log x) on Riemann Hypothesis (Cramér, 1920). log x log x Lower bound: G(x) (log x) 2 4 log3 x (F,Green,Konyagin,Maynard,Tao,2018) log2 x = log log x; log3 x = log log log x; ::: Banks, Ford, Tao Probabilistic models for primes 2 / 28 July, 2019 2 / 28 Conjectures on large prime gaps G(x) Cramér (1936): lim sup 2 = 1. x!1 log x Shanks (1964): G(x) ∼ log2 x. G(x) −γ Granville (1995): lim sup 2 > 2e = 1:1229 ::: x!1 log x G(x) Computations: sup 2 ≈ 0:92: x61018 log x Banks, Ford, Tao Probabilistic models for primes 3 / 28 July, 2019 3 / 28 Computational evidence, up to 1018 Banks, Ford, Tao Probabilistic models for primes 4 / 28 July, 2019 4 / 28 Theorem. (Cramér 1936) With probability 1, Cm+1 − Cm lim sup 2 = 1: m!1 log Cm Cramér: “for the ordinary sequence of prime numbers pn, some similar relation may hold”. Cramér’s model of large prime gaps Random set C = fC1;C2;:::g ⊂ N, choose n > 3 to be in C with probability 1 ; log n the 1/ log n matches the density of primes near n. Banks, Ford, Tao Probabilistic models for primes 5 / 28 July, 2019 5 / 28 Cramér’s model of large prime gaps Random set C = fC1;C2;:::g ⊂ N, choose n > 3 to be in C with probability 1 ; log n the 1/ log n matches the density of primes near n. Theorem. (Cramér 1936) With probability 1, Cm+1 − Cm lim sup 2 = 1: m!1 log Cm Cramér: “for the ordinary sequence of prime numbers pn, some similar relation may hold”. Banks, Ford, Tao Probabilistic models for primes 5 / 28 July, 2019 5 / 28 Cramér model and large gaps Cm+1 − Cm a.s. lim sup 2 = 1: m!1 log Cm Proof: 1 k (n + 1; : : : ; n + k 62 C) ∼ 1 − ∼ e−k/ log n: P log n k > (1 + ") log2 n, this is n−1−". Sum converges k < (1 − ") log2 n, this is n−1+". Sum diverges. Finish with Borel-Cantelli. Banks, Ford, Tao Probabilistic models for primes 6 / 28 July, 2019 6 / 28 Cramér’s model defect: global distribution Theorem. (Cramér 1936 “Probabilistic RH”)) 1/2+" With probability 1, πC(x) := #fn 6 x : n 2 Cg = li(x) + O(x ): Theorem. (Pintz) x (π (x) − li(x))2 ∼ ; E C log x Theorem. (Cramér 1920) On R.H., Z 2x 1 2 x jπ(t) − li(t)j dt 2 x x log x Banks, Ford, Tao Probabilistic models for primes 7 / 28 July, 2019 7 / 28 Theorem. (Maier 1985) π(x + logM x) − π(x) 8M > 1; lim sup M−1 > 1 x!1 log x π(x + logM x) − π(x) and lim inf < 1: x!1 logM−1 x Cramér model defect: short intervals Theorem. (Cramér model in short intervals) y With prob. 1, π (x + y) − π (x) ∼ (y/ log2 x ! 1) C C log x Theorem (Selberg). Let y ! 1. On RH, for almost all x, log2 x y π(x + y) − π(x) ∼ : log x Banks, Ford, Tao Probabilistic models for primes 8 / 28 July, 2019 8 / 28 Cramér model defect: short intervals Theorem. (Cramér model in short intervals) y With prob. 1, π (x + y) − π (x) ∼ (y/ log2 x ! 1) C C log x Theorem (Selberg). Let y ! 1. On RH, for almost all x, log2 x y π(x + y) − π(x) ∼ : log x Theorem. (Maier 1985) π(x + logM x) − π(x) 8M > 1; lim sup M−1 > 1 x!1 log x π(x + logM x) − π(x) and lim inf < 1: x!1 logM−1 x Banks, Ford, Tao Probabilistic models for primes 8 / 28 July, 2019 8 / 28 This fails for primes, e.g. H = f0; 1g, because the primes are biased For each prime p, all but one prime in 2 f1; ::; p − 1g modulo p; But C is equidistributed in f0; 1; : : : ; p − 1g mod p. Even for sets H where we expect many prime patterns, e.g. H = f0; 2g (twin primes), Cramér’s model gives the wrong prediction. Cramér’s model defect: k-correlations Theorem. (k-correlations in Cramér’s model) Let H be a finite set of integers. With probability 1, x #fn x : n + h 2 C 8h 2 Hg ∼ : 6 (log x)jHj Banks, Ford, Tao Probabilistic models for primes 9 / 28 July, 2019 9 / 28 Cramér’s model defect: k-correlations Theorem. (k-correlations in Cramér’s model) Let H be a finite set of integers. With probability 1, x #fn x : n + h 2 C 8h 2 Hg ∼ : 6 (log x)jHj This fails for primes, e.g. H = f0; 1g, because the primes are biased For each prime p, all but one prime in 2 f1; ::; p − 1g modulo p; But C is equidistributed in f0; 1; : : : ; p − 1g mod p. Even for sets H where we expect many prime patterns, e.g. H = f0; 2g (twin primes), Cramér’s model gives the wrong prediction. Banks, Ford, Tao Probabilistic models for primes 9 / 28 July, 2019 9 / 28 Hardy-Littlewood conjectures for primes x Cramér: #fn x : n + h 2 C 8h 2 Hg ∼ : 6 (log x)jHj Prime k-tuples Conjecture (Hardy-Littlewood, 1922) x #fn x : n + h prime 8h 2 Hg ∼ S(H) (x ! 1); 6 (log x)jHj where −|Hj Y jH mod pj 1 S(H) := 1 − 1 − : p p p The factor S(H) captures the bias of real primes; For each p, H must avoid the forbidden residue class 0 mod p. H is admissible if jH mod pj < p for all p. Banks, Ford, Tao Probabilistic models for primes 10 / 28 July, 2019 10 / 28 Cramér model defect: gaps Theorem: With probability 1, #fC N : C − C = kg e−k/ log N n 6 n+1 n ∼ (N ! 1) #fCn 6 Ng log N 18 Actual prime gap statistics, pn < 4 · 10 Banks, Ford, Tao Probabilistic models for primes 11 / 28 July, 2019 11 / 28 Granville’s refinement of Cramér’s model Y T = o(log x) Q = p = xo(1) p6T Real primes live in UT := fn 2 Z : gcd(n; Q) = 1g, the integers not divisible by any prime p T . The set U has density θ = Q (1 − 1/p). 6 T p6T Granville’s random model: For x < n 6 2x, choose n in G with probability ( 0 if gcd(n; Q) > 1 (i.e.,n 62 UT ) 1/θ log n if gcd(n; Q) = 1 (i.e.,n 2 UT ): k-correlations. For all H, with probability 1 we have x #fn x : n + h 2 G 8h 2 Hg ∼ S(H) ; x ! 1: 6 (log x)jHj Banks, Ford, Tao Probabilistic models for primes 12 / 28 July, 2019 12 / 28 Granville’s refinement of Cramér’s model, II UT := fn 2 Z :(n; Q) = 1g, (integers with no prime factor 6 T ) Theorem. (Granville 1995) Write G = fG1;G2;:::g. With probability 1, Gn+1 − Gn −γ lim sup 2 > 2e = 1:1229 ::: n!1 log Gn Idea: with y = c log2 x, T = y1/2+o(1), y # ([Qm; Qm + y] \U ) = # ([0; y] \U ) ∼ : T T log y By contrast, for a typical a 2 Z, Y 1 y # ([a; a + y] \U ) ∼ y 1 − ∼ 2e−γ (Mertens) T p log y p6T Banks, Ford, Tao Probabilistic models for primes 13 / 28 July, 2019 13 / 28 Minor flaw in Granville’s model Hardy-Littlewood statistics: Z x dt fn x : n + h 2 G 8h 2 Hg = S(H) + E (x; H); # 6 jHj G 2 (log t) where jHj+1 EG(x; H) = Ω(x/(log x) ): Conjecture For any admissible H, we have Z x dt fn x : n+h 8h 2 Hg = S(H) +O(x1/2+"): # 6 prime jHj 2 (log t) Much numerical evidence for this, especially for H = f0; 2g; f0; 2; 6g; f0; 4; 6g; f0; 2; 6; 8g. Banks, Ford, Tao Probabilistic models for primes 14 / 28 July, 2019 14 / 28 1 Global density: P(n 2 B) = P(n 62 Sz(n)) ∼ log n : Matches primes. Difficulty: n1 2 B, n2 2 B not independent. We conjecture that the primes and B share similar local statistics. A new “random sieve” model of primes Random set B ⊂ N: • For prime p, take a random residue class ap 2 f0; : : : ; p − 1g, uniform probability, independent for different p; • Let Sz = fn 2 Z : n 6≡ ap ( mod p); p 6 zg, random sieved set with Y e−γ density(S ) = θ = (1 − 1/p) ∼ : z z log z p6z 1/eγ 0:56::: 1 • Take z = z(n) ∼ n = n so that θz(n) ∼ log n , density of primes. • Define B = fn 2 N : n 62 Sz(n)g: Banks, Ford, Tao Probabilistic models for primes 15 / 28 July, 2019 15 / 28 A new “random sieve” model of primes Random set B ⊂ N: • For prime p, take a random residue class ap 2 f0; : : : ; p − 1g, uniform probability, independent for different p; • Let Sz = fn 2 Z : n 6≡ ap ( mod p); p 6 zg, random sieved set with Y e−γ density(S ) = θ = (1 − 1/p) ∼ : z z log z p6z 1/eγ 0:56::: 1 • Take z = z(n) ∼ n = n so that θz(n) ∼ log n , density of primes.