Introduction Large integers Distributions The universal invariant profile of the multiplicative group Greg Martin University of British Columbia joint work with Reginald M. Simpson Canadian Mathematical Society Winter Meeting Toronto, ON December 9, 2019 these slides can be found on my web page www.math.ubc.ca/ ∼ gerg/index.shtml?slides The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions Yu–Ru Liu, Stanley Xiao, and Asif Zaman Thank you for organizing this session! The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions You don’t look a day over 100 (. . . although you are) Julia Robinson (Dec. 8, 1919—July 30, 1985) Superhero of logic and computability theory, most notably for contributions to Hilbert’s 10th problem The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions Outline Introduction to the multiplicative group 1 Expected multiplicative groups for large integers 2 Distributions (somewhat) like the Erd˝ os–Kac theorem 3 The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions The multiplicative group The finite ring Z / n Z has: a cyclic additive group C n = ( Z / n Z ) + of size n ; an abelian multiplicative group M n = ( Z / n Z ) × of size φ ( n ) . Overarching question Which abelian group of φ ( n ) elements is M n ? Example: M n is cyclic if and only if n has a primitive root. Methodology—analytic number theory Choose a numerical statistic of M n , and investigate the distribution of that statistic when n is “chosen at random”. Example: Distribution of φ ( n ) known (Schoenberg, 1928). n The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions The invariant factor decomposition Two forms that answers to the question can take Primary decomposition: for G finite abelian, G ∼ = C p r 1 1 ⊕ · · · ⊕ C p k , rk where the p r j j are prime powers (unique up to reordering) Invariant factors: for G finite abelian, G ∼ = C λ 1 ⊕ · · · ⊕ C λ ℓ , where λ 1 | λ 2 | · · · | λ ℓ (unique) Another object in analytic number theory The largest invariant factor λ ℓ of M n equals the Carmichael function λ ( n ) , whose distribution has also been investigated (Erd˝ os/Pomerance/Schmutz, 1991). The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions Example: M n when n = 11 ! = 2 8 · 3 4 · 5 2 · 7 · 11 M 11 ! ∼ = M 2 8 × M 3 4 × M 5 2 × M 7 × M 11 ∼ = ( C 2 ⊕ C 64 ) ⊕ C 54 ⊕ C 20 ⊕ C 6 ⊕ C 10 ∼ = C 2 ⊕ C 64 ⊕ C 2 ⊕ C 27 ⊕ C 4 ⊕ C 5 ⊕ C 2 ⊕ C 3 ⊕ C 2 ⊕ C 5 ∼ = C 2 ⊕ C 2 ⊕ C 2 ⊕ C 2 ⊕ C 60 ⊕ C 8 , 640 64 27 4 5 5 3 2 2 2 2 64 4 2 2 2 2 8640 60 27 3 2 2 2 2 5 5 The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions Work in progress with Jenna Downey For any fixed finite abelian q -group G : an asymptotic formula for { n ≤ x : the q -Sylow subgroup of M n equals G } 20499647385305088000000 M 30 ! 55440 12 12 4 2 2 2 2 2 2 Theorem (Ben Chang–M., 2019+) The number of integers n ≤ x for which the least invariant factor of M n does not equal 2 is ∼ Cx / √ log x for a certain C > 0 . The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions Connection to ω ( n ) Theorem (Erd˝ os–Kac theorem) The limiting distribution of the normalized statistic ( number of prime factors of n ) − log log n ( log log n ) 1 / 2 is the standard normal random variable. 0.4 0.3 0.2 0.1 - 4 - 2 2 4 The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions Connection to ω ( n ) Theorem (Erd˝ os–Kac theorem) The limiting distribution of the normalized statistic ( number of prime factors of n ) − log log n ( log log n ) 1 / 2 is the standard normal random variable. Theorem (M.–Lee Troupe 2018, answering a question of Vukoslavcevic and Shparlinski) For certain constants A , B > 0 , the limiting distribution of log ( number of subgroups of M n ) − A ( log log n ) 2 ( B ( log log n ) 3 ) 1 / 2 is the standard normal random variable. The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions M n for a random integer n near e e 1 , 000 , 000 The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions M n for a random integer n near e e 1 , 000 , 000 2 ( multiplicity 494,790 ) 12 ( multiplicity 242,988 ) 120 ( multiplicity 76,400 ) 2,520 ( multiplicity 48,092 ) 5,040 ( multiplicity 19,524 ) 55,440 ( multiplicity 16,133 ) 720,720 ( multiplicity 23,001 ) 24,504,480 ( multiplicity 2,662 ) The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions M n for a random integer n near e e 1 , 000 , 000 2 ( multiplicity 494,790 ) 12 ( multiplicity 242,988 ) 120 ( multiplicity 76,400 ) 6 ( multiplicity 624 ) 2,520 ( multiplicity 48,092 ) 60 ( multiplicity 1,103 ) 5,040 ( multiplicity 19,524 ) 840 ( multiplicity 111 ) 1,441,440 ( multiplicity 100 ) 55,440 ( multiplicity 16,133 ) 720,720 ( multiplicity 23,001 ) 24,504,480 ( multiplicity 2,662 ) The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions M n for a random integer n near e e 1 , 000 , 000 2 ( proportion ≈ 1 / 2 ) 12 ( proportion ≈ 1 / 4 ) 120 ( proportion ≈ 1 / 12 ) 2,520 ( proportion ≈ 1 / 24 ) 5,040 ( proportion ≈ 1 / 40 ) 55,440 ( proportion ≈ 1 / 60 ) 720,720 ( proportion ≈ 1 / 48 ) 24,504,480 ( proportion ≈ 1 / 144 ) The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions That integer wasn’t so special after all Theorem (M.–Reginald M. Simpson, > 2019 ) For almost all integers n , the multiplicative group M n has: ∼ 1 2 log log n invariant factors equal to 2 ; ∼ 1 4 log log n invariant factors equal to 12 ; ∼ 1 12 log log n invariant factors equal to 120 ; ∼ 1 24 log log n invariant factors equal to 2 , 520 ; ∼ 1 40 log log n invariant factors equal to 5 , 040 ; ∼ 1 60 log log n invariant factors equal to 55 , 440 ; ∼ 1 48 log log n invariant factors equal to 720 , 720 ; . . . Interpretation: The important structure arithmetic modulo n seems to be encoded almost completely in the largest few invariant factors. The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions What are those sequences of numbers? Definition: Prime-power totients { φ ( p r ): p prime , r ≥ 1 } = { 1 , 2 , 4 , 6 , 8 , 10 , 12 , 16 , 18 , 20 , . . . } = { ppt 1 , ppt 2 , ppt 3 , . . . } Definition: Cumulative least common multiples λ ( x ) = lcm [ p r : φ ( p r ) ≤ x ] , λ k = λ ( ppt k ) Theorem (M.–Reginald M. Simpson, > 2019 ) For almost all integers n , the multiplicative group M n has � 1 � 1 ∼ − log log n ppt k ppt k + 1 invariant factors equal to λ k for each k = 1 , 2 , 3 , . . . . The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions Example invariant factor: 55 , 440 We can be more precise than ∼ 1 60 log log n about the multiplicity of the invariant factor 55 , 440 : Theorem (M.–Simpson, > 2019 ) The limiting distribution of the normalized count ( multiplicity of the invariant factor 55 , 440) − 1 60 log log n ( 1 6 log log n ) 1 / 2 is the standard normal random variable. 0.4 0.3 0.2 0.1 - 4 - 2 2 4 The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions Example invariant factor: 2 (but lying) Theorem (M.–Simpson, > 2019 ) The limiting distribution of the normalized count ( multiplicity of the invariant factor 2) − 1 2 log log n ( 1 2 log log n ) 1 / 2 is the standard normal random variable. 0.4 0.3 0.2 0.1 - 4 - 2 2 4 The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions Example invariant factor: 2 (the truth) Theorem (M.–Simpson, > 2019 ) The limiting distribution of the normalized count ( multiplicity of the invariant factor 2) − 1 2 log log n ( 1 2 log log n ) 1 / 2 1 is a skew-normal random variable with “shape” parameter 3 . √ 0.4 0.3 0.2 0.1 - 4 - 2 2 4 The universal invariant profile of the multiplicative group Greg Martin
Introduction Large integers Distributions Sylow- 2 and - 3 subgroups for M 101 ! 26588814358957503287787 3 3 3 3 3 3 3 3 9 9 9 39614081257132168796771975168 32 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 816 101 ! has 26 prime factors . . . 12 of them are ≡ 1 (mod 4) , and 11 of them are ≡ 1 (mod 3) The universal invariant profile of the multiplicative group Greg Martin
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