Probabilistic models for primes and large gaps William Banks Kevin - PowerPoint PPT Presentation

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 p n � x ( p n − p n − 1 ) , p n is the n th prime. Def: G ( x ) = max 2 , 3 , 5 , 7 , . . . , 109 , 113 , 127 , 131 , . . . , 9547 , 9551 , 9587 , 9601 , . . . Upper bound: G ( x ) = O ( x 0 . 525 ) (Baker-Harman-Pintz, 2001). Improve to O ( x 1/2 log x ) on Riemann Hypothesis (Cramér, 1920). Lower bound: G ( x ) ≫ ( log x ) log 2 x log 4 x log 3 x (F,Green,Konyagin,Maynard,Tao,2018) log 2 x = log log x, log 3 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 log 2 x = 1 . x →∞ Shanks (1964) : G ( x ) ∼ log 2 x . G ( x ) log 2 x � 2 e − γ = 1 . 1229 . . . Granville (1995): lim sup x →∞ G ( x ) Computations: sup log 2 x ≈ 0 . 92 . x � 10 18 Banks, Ford, Tao Probabilistic models for primes 3 / 28 July, 2019 3 / 28

Computational evidence, up to 10 18 Banks, Ford, Tao Probabilistic models for primes 4 / 28 July, 2019 4 / 28

Theorem. (Cramér 1936) With probability 1, lim sup log Cramér: “ for the ordinary sequence of prime numbers , some similar relation may hold ”. Cramér’s model of large prime gaps Random set C = { C 1 , C 2 , . . . } ⊂ 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 = { C 1 , C 2 , . . . } ⊂ 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, C m +1 − C m lim sup = 1 . log 2 C m m →∞ Cramér: “ for the ordinary sequence of prime numbers p n , some similar relation may hold ”. Banks, Ford, Tao Probabilistic models for primes 5 / 28 July, 2019 5 / 28

Cramér model and large gaps C m +1 − C m a.s. lim sup = 1 . log 2 C m m →∞ Proof: � � k 1 ∼ e − k / log n . P ( n + 1 , . . . , n + k �∈ C ) ∼ 1 − log n k > (1 + ε ) log 2 n , this is ≪ n − 1 − ε . Sum converges k < (1 − ε ) log 2 n , this is ≫ n − 1+ ε . 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”)) With probability 1, π C ( x ) := # { n � x : n ∈ C } = li ( x ) + O ( x 1/2+ ε ) . Theorem. (Pintz) x E ( π C ( x ) − li ( x )) 2 ∼ log x, Theorem. (Cramér 1920) On R.H., � 2 x 1 x | π ( t ) − li ( t ) | 2 dt ≪ log 2 x x x Banks, Ford, Tao Probabilistic models for primes 7 / 28 July, 2019 7 / 28

Theorem. (Maier 1985) log lim sup log log and lim inf log Cramér model defect: short intervals Theorem. (Cramér model in short intervals) y ( y / log 2 x → ∞ ) With prob. 1, π C ( x + y ) − π C ( x ) ∼ log x y Theorem (Selberg) . Let log 2 x → ∞ . On RH, for almost all 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 ( y / log 2 x → ∞ ) With prob. 1, π C ( x + y ) − π C ( x ) ∼ log x y Theorem (Selberg) . Let log 2 x → ∞ . On RH, for almost all x , y π ( x + y ) − π ( x ) ∼ log x. Theorem. (Maier 1985) π ( x + log M x ) − π ( x ) lim sup ∀ M > 1 , > 1 log M − 1 x x →∞ π ( x + log M x ) − π ( x ) and lim inf < 1 . log M − 1 x x →∞ Banks, Ford, Tao Probabilistic models for primes 8 / 28 July, 2019 8 / 28

This fails for primes, e.g. , because the primes are biased For each prime , all but one prime in modulo ; But is equidistributed in mod . Even for sets where we expect many prime patterns, e.g. (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 # { n � x : n + h ∈ C ∀ h ∈ H} ∼ ( log x ) |H| . 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 # { n � x : n + h ∈ C ∀ h ∈ H} ∼ ( log x ) |H| . This fails for primes, e.g. H = { 0 , 1 } , because the primes are biased For each prime p , all but one prime in ∈ { 1 , .., p − 1 } modulo p ; But C is equidistributed in { 0 , 1 , . . . , p − 1 } mod p . Even for sets H where we expect many prime patterns, e.g. H = { 0 , 2 } (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: # { n � x : n + h ∈ C ∀ h ∈ H} ∼ ( log x ) |H| . Prime k -tuples Conjecture (Hardy-Littlewood, 1922) x # { n � x : n + h prime ∀ h ∈ H} ∼ S ( H ) ( x → ∞ ) , ( log x ) |H| where � �� � −|H| 1 − |H mod p | 1 − 1 � S ( H ) := . 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 |H mod p | < 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, ∼ e − k / log N # { C n � N : C n +1 − C n = k } ( N → ∞ ) # { C n � N } log N Actual prime gap statistics, p n < 4 · 10 18 Banks, Ford, Tao Probabilistic models for primes 11 / 28 July, 2019 11 / 28

Granville’s refinement of Cramér’s model � p = x o (1) T = o ( log x ) Q = p � T Real primes live in U T := { n ∈ Z : gcd ( n, Q ) = 1 } , the integers not divisible by any prime p � T . The set U T has density θ = � p � T (1 − 1/ p ) . Granville’s random model: For x < n � 2 x , choose n in G with probability � if gcd ( n, Q ) > 1 ( i.e., n �∈ U T ) 0 1/ θ if gcd ( n, Q ) = 1 ( i.e., n ∈ U T ) . log n k -correlations. For all H , with probability 1 we have x # { n � x : n + h ∈ G ∀ h ∈ H} ∼ S ( H ) ( log x ) |H| , x → ∞ . Banks, Ford, Tao Probabilistic models for primes 12 / 28 July, 2019 12 / 28

Granville’s refinement of Cramér’s model, II U T := { n ∈ Z : ( n, Q ) = 1 } , (integers with no prime factor � T ) Theorem. (Granville 1995) Write G = { G 1 , G 2 , . . . } . With probability 1, G n +1 − G n � 2 e − γ = 1 . 1229 . . . lim sup log 2 G n n →∞ Idea : with y = c log 2 x , T = y 1/2+ o (1) , y # ([ Qm, Qm + y ] ∩ U T ) = # ([0 , y ] ∩ U T ) ∼ log y. By contrast, for a typical a ∈ Z , � � 1 − 1 y � ∼ 2 e − γ # ([ a, a + y ] ∩ U T ) ∼ y (Mertens) log y p p � T Banks, Ford, Tao Probabilistic models for primes 13 / 28 July, 2019 13 / 28

Minor flaw in Granville’s model Hardy-Littlewood statistics: � x dt # { n � x : n + h ∈ G ∀ h ∈ H} = S ( H ) ( log t ) |H| + E G ( x ; H ) , 2 where E G ( x ; H ) = Ω( x /( log x ) |H| +1 ) . Conjecture For any admissible H , we have � x dt # { n � x : n + h prime ∀ h ∈ H} = S ( H ) ( log t ) |H| + O ( x 1/2+ ε ) . 2 Much numerical evidence for this, especially for H = { 0 , 2 } , { 0 , 2 , 6 } , { 0 , 4 , 6 } , { 0 , 2 , 6 , 8 } . Banks, Ford, Tao Probabilistic models for primes 14 / 28 July, 2019 14 / 28

Global density: Matches primes. log Difficulty: , not independent. We conjecture that the primes and share similar local statistics . A new “random sieve” model of primes Random set B ⊂ N : • For prime p , take a random residue class a p ∈ { 0 , . . . , p − 1 } , uniform probability, independent for different p ; • Let S z = { n ∈ Z : n �≡ a p ( mod p ) , p � z } , random sieved set with (1 − 1/ p ) ∼ e − γ � density ( S z ) = θ z = log z . p � z • Take z = z ( n ) ∼ n 1/ e γ = n 0 . 56 ... so that θ z ( n ) ∼ 1 log n , density of primes. • Define B = { n ∈ N : n �∈ S z ( n ) } . 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 a p ∈ { 0 , . . . , p − 1 } , uniform probability, independent for different p ; • Let S z = { n ∈ Z : n �≡ a p ( mod p ) , p � z } , random sieved set with (1 − 1/ p ) ∼ e − γ � density ( S z ) = θ z = log z . p � z • Take z = z ( n ) ∼ n 1/ e γ = n 0 . 56 ... so that θ z ( n ) ∼ 1 log n , density of primes. • Define B = { n ∈ N : n �∈ S z ( n ) } . 1 Global density: P ( n ∈ B ) = P ( n �∈ S z ( n ) ) ∼ log n . Matches primes. Difficulty: n 1 ∈ B , n 2 ∈ B not independent. We conjecture that the primes and B share similar local statistics . Banks, Ford, Tao Probabilistic models for primes 15 / 28 July, 2019 15 / 28