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Deshouillers–Dress–Tenenbaum theorem

The Deshouillers–Dress–Tenenbaum theorem (or in short DDT theorem) is a result from probabilistic number theory, which describes the probability distribution of a divisor d {\displaystyle d} of a natural number n {\displaystyle n} within the interval [ 1 , n ] {\displaystyle [1,n]} , where the divisor d {\displaystyle d} is chosen uniformly. More precisely, the theorem deals with the sum of distribution functions of the logarithmic ratio of divisors to growing intervals. The theorem states that the Cesàro sum of the distribution functions converges to the arcsine distribution, meaning that small and large values have a high probability. The result is therefor also referred to as the arcsine law of Deshouillers–Dress–Tenenbaum.

The theorem was proven in 1979 by the French mathematicians Jean-Marc Deshouillers, François Dress, and Gérald Tenenbaum. The result was generalized in 2007 by Gintautas Bareikis and Eugenijus Manstavičius.

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Deshouillers–Dress–Tenenbaum theorem

Let n ≥ 1 {\displaystyle n\geq 1} be a natural number and fix the following notation:

  • T ( n , r ) = { s ∈ N : s | n , ; s ≤ r } {\displaystyle T(n,r)=\{s\in \mathbb {N} :s|n,;s\leq r\}} is the set of divisors of n {\displaystyle n} that are smaller or equal than r {\displaystyle r} .
  • τ ( n , r ) := | T ( n , r ) | {\displaystyle \tau (n,r):=|T(n,r)|} is the number of divisors of n {\displaystyle n} that are smaller or equal than r {\displaystyle r} .
  • T ( n ) := T ( n , n ) {\displaystyle T(n):=T(n,n)}
  • τ ( n ) := | T ( n , n ) | {\displaystyle \tau (n):=|T(n,n)|}
  • ( Ω , A , P ) {\displaystyle (\Omega ,{\mathcal {A}},P)} is a probability space.

Introduction

Let d : Ω → T ( n ) {\displaystyle d:\Omega \to T(n)} be a uniformly distributed random variable on the set of divisors of n {\displaystyle n} and consider the logarithmic ratio

D n := log ⁡ ( d ) log ⁡ ( n ) {\displaystyle D_{n}:={\frac {\log(d)}{\log(n)}}} ,

notice that the realizations of the random variable D n {\displaystyle D_{n}} are characterized entirely by the divisors of n {\displaystyle n} and each divisor has probability 1 / τ ( n ) {\displaystyle 1/\tau (n)} . The distribution function of D n {\displaystyle D_{n}} is defined as

P ( D n ≤ t ) := 1 τ ( n ) ∑ s | n , s ≤ n t 1 = τ ( n , n t ) τ ( n ) , {\displaystyle \mathbb {P} (D_{n}\leq t):={\frac {1}{\tau (n)}}\sum \limits _{s|n,s\leq n^{t}}1={\frac {\tau (n,n^{t})}{\tau (n)}},\quad } for 0 ≤ t ≤ 1 {\displaystyle 0\leq t\leq 1} .

It is easy to see that the sequence D 1 , D 2 , … , D n , … {\displaystyle D_{1},D_{2},\dots ,D_{n},\dots } does not converge in distribution when considering subsequences indexed by prime numbers D p 1 , D p 2 , … {\displaystyle D_{p_{1}},D_{p_{2}},\dots } therefore one is interested in the Césaro sum.3

Statement

Let ( D n ) n ≥ 1 {\displaystyle (D_{n})_{n\geq 1}} be a sequence of the above-defined random variables and let x ≥ 2 {\displaystyle x\geq 2} . Then for all t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} the Cesàro mean satisfies uniform convergence to

1 x ∑ n ≤ x P ( D n ≤ t ) = 2 π arcsin ⁡ t + O ( 1 log ⁡ ( x ) ) {\displaystyle {\frac {1}{x}}\sum \limits _{n\leq x}\mathbb {P} (D_{n}\leq t)={\frac {2}{\pi }}\arcsin {\sqrt {t}}+{\mathcal {O}}\left({\frac {1}{\sqrt {\log(x)}}}\right)} .4

Further Results

Eugenijus Manstavičius, Gintautas Bareikis, and Nikolai Timofeev extended the theorem by replacing the counting function 1 {\displaystyle 1} in τ ( n , v ) {\displaystyle \tau (n,v)} with a multiplicative function f : N → R + {\displaystyle f:\mathbb {N} \to \mathbb {R} _{+}} and studied the stochastic behavior of

X ( n , v ) := M ( n , v ) M ( n ) {\displaystyle X(n,v):={\frac {M(n,v)}{M(n)}}} ,

where

M ( n , v ) := ∑ s | n , s ≤ v f ( s ) , M ( n ) := M ( n , n ) {\displaystyle M(n,v):=\sum \limits _{s|n,s\leq v}f(s),\quad M(n):=M(n,n)} .

Result of Manstavičius-Timofeev

Let D [ 0 , 1 ] {\displaystyle \mathbb {D} [0,1]} be the Skorokhod space and let B ( D [ 0 , 1 ] ) {\displaystyle {\mathcal {B}}(\mathbb {D} [0,1])} be the Borel σ-algebra. For 1 ≤ m ≤ x {\displaystyle 1\leq m\leq x} , define a discrete measure μ x ( { m } ) := 1 / [ x ] {\displaystyle \mu _{x}(\{m\}):=1/[x]} , describing the probability of selecting m {\displaystyle m} from [ 1 , x ] {\displaystyle [1,x]} with probability 1 / [ x ] {\displaystyle 1/[x]} .

Manstavičius and Timofeev studied the process ( X x ) x ≥ m {\displaystyle \left(X_{x}\right)_{x\geq m}} with

X x := X x ( n , t ) = M ( n , x t ) M ( n ) {\displaystyle X_{x}:=X_{x}(n,t)={\frac {M(n,x^{t})}{M(n)}}}

for t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} and the image measure μ x ∘ X x − 1 {\displaystyle \mu _{x}\circ X_{x}^{-1}} on D [ 0 , 1 ] {\displaystyle \mathbb {D} [0,1]} .

That is, the image measure is defined for B ∈ B ( D [ 0 , 1 ] ) {\displaystyle B\in {\mathcal {B}}(\mathbb {D} [0,1])} as follows:

μ x ( B ) := 1 [ x ] ∑ m ≤ x 1 B ( X x ( m , ⋅ ) ) . {\displaystyle \mu _{x}(B):={\frac {1}{[x]}}\sum \limits _{m\leq x}1_{B}(X_{x}(m,\cdot )).} 5

They showed that if f ( p ) = C > 1 {\displaystyle f(p)=C>1} for every prime number p {\displaystyle p} and f ( p k ) ≥ 0 {\displaystyle f(p^{k})\geq 0} for all prime numbers p {\displaystyle p} and all k ≥ 2 {\displaystyle k\geq 2} , then μ x ∘ X x − 1 {\displaystyle \mu _{x}\circ X_{x}^{-1}} converges weakly to a measure in D [ 0 , 1 ] {\displaystyle \mathbb {D} [0,1]} as x → ∞ {\displaystyle x\to \infty } .6

Result of Bareikis-Manstavičius

Bareikis and Manstavičius generalized the theorem of Deshouillers-Dress-Tenenbaum and derived a limit theorem for the sum

S x ( t ) := 1 x ∑ m ≤ x M ( m , m t ) M ( m ) {\displaystyle S_{x}(t):={\frac {1}{x}}\sum \limits _{m\leq x}{\frac {M(m,m^{t})}{M(m)}}}

for a class of multiplicative functions f {\displaystyle f} that satisfy certain analytical properties. The resulting distribution is the more general beta distribution.7

References

  1. Deshouillers, Jean-Marc; Dress, François; Tenenbaum, Gérald (1979). "Lois de répartition des diviseurs, 1". Acta Arithmetica (in French). 34 (4): 273–283. http://eudml.org/doc/205609 ↩

  2. Bareikis, Gintautas; Manstavičius, Eugenijus (2007). "On the DDT theorem". Acta Arithmetica. 126 (2): 155–168. http://eudml.org/doc/278080 ↩

  3. Deshouillers, Jean-Marc; Dress, François; Tenenbaum, Gérald (1979). "Lois de répartition des diviseurs, 1". Acta Arithmetica (in French). 34 (4): 273–283. http://eudml.org/doc/205609 ↩

  4. Deshouillers, Jean-Marc; Dress, François; Tenenbaum, Gérald (1979). "Lois de répartition des diviseurs, 1". Acta Arithmetica (in French). 34 (4): 274. http://eudml.org/doc/205609 ↩

  5. Eugenijus Manstavičius and Nikolai Mikhailovich Timofeev (1997). "A functional limit theorem related to natural divisors". Acta Mathematica Hungarica. 75: 1–13. doi:10.1023/A:1006501331306. /wiki/Doi_(identifier) ↩

  6. Bareikis, Gintautas; Manstavičius, Eugenijus (2007). "On the DDT theorem". Acta Arithmetica. 126 (2): 155–168. http://eudml.org/doc/278080 ↩

  7. Bareikis, Gintautas; Manstavičius, Eugenijus (2007). "On the DDT theorem". Acta Arithmetica. 126 (2): 155–168. http://eudml.org/doc/278080 ↩