Large gaps in sets of primes and other sequences I. Heuristics and basic constructions Kevin Ford University of Illinois at Urbana-Champaign May, 2018 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) x0:525 (Baker-Harman-Pintz, 2001). Improve to O(x1/2+") on RH. log2 x log4 x Lower bound: G(x) (log x) (F,Green,Konyagin,Maynard,Tao,2018) log3 x Conjectures 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 Computational evidence, up to 1018 Cramér’s model of large prime gaps Let X3;X4;X5;::: be indep. random vars. s.t. 1 1 (X = 1) = ; (X = 0) = 1 − : P n log n P n log n Let P = fn : Xn = 1g = fP1;P2;:::g; the set of “probabilistic primes”. Theorem (Cramér, 1936) With probability 1, PN+1 − PN lim sup 2 = 1: N!1 log N Cramér: “for the ordinary sequence of prime numbers pn, some similar relation may hold”. Sketch of the proof of Cramér’s theorem 1 1 (X = 1) = ; (X = 0) = 1 − : P n log n P n log n 1−o(1) Suppose that N 6 k 6 N. Then 1 g (X = ··· = X = 0) ≈ 1 − ≈ e−g/ log N : P k+1 k+g log k Hence (summing on k) −g/ log N E#fgaps of length > g below Ng ≈ Ne : If g > (1 + ") log2 N, this is o(1). If g < (1 − ") log2 N, then this is very large. More predictions of Cramér’s model: distribtion of gaps −λ (1/N)E#fgaps of length > λ log Ng ≈ e : 18 Actual prime gap statistics, pn < 4 · 10 Gallagher, 1976. Prime k−tuples conjecture ) exponential prime gap distribution General Cramér’s-type model: random darts Choose N random points in [0; 1] (random darts) Theorem (classical?) log N W.h.p., the max. gap is ∼ N . Proof idea (Rényi). The N + 1 gaps have distribution E E =d 1 ;:::; N+1 ;S = E + ··· + E ; S S 1 N+1 −x where each Ei has exponential distribution, P(Ei 6 x) = 1 − e . W.h.p., S ∼ N. Also, e−u N+1 (max E log N + u) = 1 − ∼ exp{−e−ug; P i 6 N the Gumbel extreme value distribution. Random darts and Cramér’s model Choose N random points in [0; x] (random darts) x(log N + u) max gap ≈ exp{−e−ug: P 6 N Probabilistic primes; N = li(x) + O(x1/2+") x(log li(x) + u) −u P max Pn+1 − Pn 6 ≈ exp{−e g: Pn6x li(x) Theorem For Cramér’s probabilistic primes, x log li(x) max Pn − Pn−1 . Cr1(x) := ≈ (log x)(log x − log2 x): Pn6x li(x) Q1: Does this explain the data for actual primes? Data vs. refined Cramér conjecture General Cramér’s-type model: random darts Choose N random points in [0; x] (random darts) x log N Expected maximal gap is of size ≈ N Prime k-tuples. Let f1; : : : ; fk be distinct, irreducible polynomials fi : Z ! Z with pos. leading coeff., degrees di, and f1 ··· fk has no fixed prime factor. Conjecture (Bateman-Horn) #fn 6 x : f1(n); : : : ; fk(n) all primeg ∼ C lik(x); R x dt where C = C(f1; : : : ; fk) > 0 is constant and lik(x) = 2 (log t)k Refined Cramér conjecture The largest gap in fn 6 x : f1(n); : : : ; fk(n) all primeg is k x log(C lik(x)) (log x) . Crk(x) := ≈ (log x − k log2 x) C lik(x) C Twin prime gaps (f1; f2) = (n; n + 2), C ≈ 1:32032 Prime triplet gaps (f1; f2; f3) = (n; n + 2; n + 6) C ≈ 2:85825 Prime quadruplet gaps (f1; : : : ; f4) = (n; n + 2; n + 6; n + 8) C ≈ 4:15118 Cramér’s model defect: global distribution of primes Theorem (Cramér, 1936 (“Probabilistic RH”)) 1/2+" With probability 1, Π(x) := #fPn 6 xg = li(x) + O(x ): J. Pintz observed the following: Theorem x (Π(x) − li(x))2 ∼ ; E log x contrast with Theorem (Cramér,1920) On R.H., Z 2x 1 2 x jπ(t) − li(t)j dt 2 x x log x Cramér’s model defect: small gaps Theorem. With probability 1, #fn : n; n + 1 2 Pg = 1 This does not hold for real primes! Theorem. With probability 1, x #fn 6 x : n; n + 2 2 Pg ∼ : log2 x Conjecture (Hardy-Littlewood, 1923). x #fn 6 x : n; n + 2 primeg ∼ C ; log2 x Q 2 where C = 2 p>2(1 − 1/(p − 1) ) ≈ 1:3203 Granville’s refinement of Cramér’s model Cramér’s model major defect: Cramér primes are equidistributed modulo small primes like 2,3,..., whereas real primes are not. This shows up in asymptotics for prime k-tuples, and for counts of primes in very short intervals: w.h.p., y Π(x + y) − Π(x) ∼ (y/ log2 x ! 1) log x By contrast, Theorem (H. Maier, 1985) 8M > 1, π(x + logM x) − π(x) lim sup M−1 > 1: x!1 log x Granville’s refinement of Cramér’s model, II Y o(1) Let T = " log x; QT = p = x : p6T Real primes live in ST := fn 2 Z :(n; QT ) = 1g For n 2 ST \ (x; 2x], define the random variables Zn : P(Zn = 1) = θ/ log n; P(Zn = 0) = 1 − θ/ log n; −γ where 1/θ = φ(QT )/QT ∼ e / log T is the density of ST . That is, θ = conditional prob. that n is prime given that n 2 S : log n T Granville’s refinement of Cramér’s model, III ST := fn 2 Z :(n; QT ) = 1g Let y = c log2 x, take special values of m 2 (x; 2x], namely those 2+o(1) with with QT jm. Since y = T , y # ([m; m + y] \S ) = # ([0; y] \S ) ∼ : (sp) T T log y By contrast, for a typical m 2 Z, y # ([m; m + y] \S ) ∼ θ−1y ∼ 2e−γ : (ty) T log y Note 2e−γ = 1:1229 : : : > 1, so the intervals in (sp) are deficient in sifted numbers. Granville’s refinement of Cramér’s model, IV T = " log x, y = c log2 x, y # ([m; m + y] \S ) = # ([0; y] \S ) ∼ : (sp) T T log y Get y/ log y P Zn = 0 : n 2 [m; m + y] \ST ≈ (1 − θ/ log x) γ ≈ e−c(e /2) log x: −γ 2 Therefore, gaps of size > (2e + o(1))(log x) exist w.h.p. Computing secondary terms; get gaps of size (log x)2 2e−γ(log x)2 + A(") + ··· ;A(") ! 1(" ! 0): log2 x Project: work out the secondary term; compare with data. Proving large gaps: Jacobsthal’s function S = fn 2 :(n; Q ) = 1g;Q = Q p: T Z T T p6T Main goal: Find J(T ), the largest gap in ST . G(2QT ) > J(T );G(x) := max pn+1 − pn: pn6x T Since QT ≈ e , get G(x) & J(log x): γ Trivial: Avg. gap is ∼ e log T ;J(T ) > T − 2 ([2;T ] \ST = ;) Lower bound (FGKMT, 2018). J(T ) T log T log3 T . log2 T Upper bound (Iwaniec, 1978). J(T ) T 2(log T )2. Conjecture (Maier-Pomerance, 1990). J(T ) = T (log T )2+o(1). Random dart model prediction: J(T ) ∼ T QT ∼ eγT log T . φ(QT ) Finding large gaps in ST Covering: J(T ) is the largest y so that there are a2; a3; a5;::: with fap mod p : p 6 T g ⊇ [0; y] Classical 3-stage-process (Westzynthius’s-Erdős-Rankin) 1 Take ap = 0 for p 2 (z; x/2] \ [2; 2y/x]. Uncovered: z-smooth numbers (few for appropriate z) and primes; Total ∼ y/ log y numbers uncovered.Far better that typical choice, which leaves about y Q (1 − 1/p) ∼ y log x uncovered z<p6x/2 log z numbers 2 Greedy choice for ap, p 2 (2y/x; z]; Unconvered: log z x . (y/ log y) log(2y/x) numbers. Want this to be 6 4 log x . 3 use each ap for p 2 (x/2; x] to cover the remaining uncovered elements of [1; y], one element for each p. x If fewer than π(x) − π(x/2) ∼ 2 log x elements left after stages 1-2, then succeed! Lower bounds for J(T ): prime k-tuples Lower bound (FGKMT, 2018). J(T ) T log T log3 T . log2 T Conjecture (Maier-Pomerance, 1990). J(T ) = T (log T )2+o(1). Random dart model prediction: J(T ) ∼ T QT ∼ eγT log T . φ(QT ) Maier-Pomerance: In Step 3, show that many ap mod p can cover two remining elements; uses “twin-prime on average” results. FGKMT: Use new prime detecting sieve (GPY-Maynard-Tao) to find ap mod p which cover many remaining elements. Assuming a uniform H-L prime k-tuples conjecture: Cover even more remaining elements with ap’s. Improve lower bound to J(T ) T (log T )1+c. Open Problems 1 Select a residue ap 2 Z/pZ for each p 6 x, let [ S = [0; x] n (ap mod p): p6x I. When ap = 0 for all p, jSj = 1 (extremal case). II. A random choice yields jSj ∼ x(e−γ/ log x).