View source for 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]] <math>d</math> of a [[natural number]] <math>n</math> within the interval <math>[1,n]</math>, where the divisor <math>d</math> is chosen [[uniform distribution (discrete)|uniformly]]. More precisely, the theorem deals with the sum of distribution functions of the [[logarithm|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 [[France|French]] mathematicians [[Jean-Marc Deshouillers]], [[François Dress]], and [[Gérald Tenenbaum]].<ref name="DDT">{{cite journal |first1=Jean-Marc |last1=Deshouillers |first2=François |last2=Dress |first3=Gérald |last3=Tenenbaum |title=Lois de répartition des diviseurs, 1 |journal=[[Acta Arithmetica]] |volume=34 |issue=4 |year=1979 |pages=273–283 |language=fr |url=http://eudml.org/doc/205609}}</ref> The result was generalized in 2007 by [[Gintautas Bareikis]] and [[Eugenijus Manstavičius]].<ref name="BM">{{cite journal |first1=Gintautas |last1=Bareikis |first2=Eugenijus |last2=Manstavičius |title=On the DDT theorem |journal=[[Acta Arithmetica]] |volume=126 |issue=2 |year=2007 |pages=155–168 |url=http://eudml.org/doc/278080}}</ref>

==Deshouillers–Dress–Tenenbaum theorem==
Let <math>n\geq 1</math> be a natural number and fix the following notation:
*<math>T(n,r)=\{s\in \mathbb{N}:s|n,;s\leq r\}</math> is the set of divisors of <math>n</math> that are smaller or equal than <math>r</math>.
*<math>\tau(n,r):=|T(n,r)|</math> is the number of divisors of <math>n</math> that are smaller or equal than <math>r</math>.
*<math>T(n):=T(n,n)</math>
*<math>\tau(n):=|T(n,n)|</math>
*<math>(\Omega,\mathcal{A},P)</math> is a [[probability space]].

===Introduction===
Let <math>d:\Omega \to T(n)</math> be a [[uniform distribution (discrete)|uniformly distributed]] random variable on the set of divisors of <math>n</math> and consider the logarithmic ratio
:<math>D_n:=\frac{\log(d)}{\log(n)}</math>,
notice that the realizations of the random variable <math>D_n</math> are characterized entirely by the divisors of <math>n</math> and each divisor has probability <math>1/\tau(n)</math>. The distribution function of <math>D_n</math> is defined as
:<math>\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</math> for <math>0\leq t \leq 1</math>.

It is easy to see that the sequence <math>D_1,D_2,\dots,D_n,\dots</math> does not [[convergence in distribution|converge in distribution]] when considering subsequences indexed by [[prime number]]s <math>D_{p_1},D_{p_2},\dots</math> therefore one is interested in the Césaro sum.<ref name="DDT" />

===Statement===
Let <math>(D_n)_{n\geq 1}</math> be a sequence of the above-defined random variables and let <math>x\geq 2</math>. Then for all <math>t\in[0,1]</math> the [[Cesàro mean]] satisfies [[uniform convergence]] to
:<math>\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)</math>.<ref>{{cite journal |first1=Jean-Marc |last1=Deshouillers |first2=François |last2=Dress |first3=Gérald |last3=Tenenbaum |title=Lois de répartition des diviseurs, 1 |journal=[[Acta Arithmetica]] |volume=34 |issue=4 |year=1979 |pages=274 |language=fr |url=http://eudml.org/doc/205609}}</ref>

==Further Results==
Eugenijus Manstavičius, Gintautas Bareikis, and [[Nikolai Mikhailovich Timofeev|Nikolai Timofeev]] extended the theorem by replacing the counting function <math>1</math> in <math>\tau(n,v)</math> with a [[multiplicative function]] <math>f:\mathbb{N}\to \mathbb{R}_+</math> and studied the stochastic behavior of
:<math>X(n,v):=\frac{M(n,v)}{M(n)}</math>,
where
:<math>M(n,v):=\sum\limits_{s|n, s\leq v}f(s), \quad M(n):=M(n,n)</math>.

=== Result of Manstavičius-Timofeev ===
Let <math>\mathbb{D}[0,1]</math> be the [[Skorokhod space]] and let <math>\mathcal{B}(\mathbb{D}[0,1])</math> be the [[Borel σ-algebra]]. For <math>1\leq m\leq x</math>, define a discrete measure <math>\mu_x(\{m\}):=1/[x]</math>, describing the probability of selecting <math>m</math> from <math>[1,x]</math> with probability <math>1/[x]</math>.

Manstavičius and Timofeev studied the process <math>\left(X_x\right)_{x\geq m}</math> with
:<math>X_x:=X_x(n,t)=\frac{M(n,x^t)}{M(n)}</math>
for <math>t\in [0,1]</math> and the image measure <math>\mu_x\circ X_x^{-1}</math> on <math>\mathbb{D}[0,1]</math>.

That is, the image measure is defined for <math>B\in \mathcal{B}(\mathbb{D}[0,1])</math> as follows:
:<math>\mu_x(B):=\frac{1}{[x]}\sum\limits_{m\leq x} 1_{B}(X_x(m,\cdot)).</math><ref>{{Cite journal |author=Eugenijus Manstavičius and Nikolai Mikhailovich Timofeev |title=A functional limit theorem related to natural divisors |journal=Acta Mathematica Hungarica |volume=75 |year=1997 |pages=1–13 |doi=10.1023/A:1006501331306}}</ref>

They showed that if <math>f(p)=C>1</math> for every prime number <math>p</math> and <math>f(p^k)\geq 0</math> for all prime numbers <math>p</math> and all <math>k\geq 2</math>, then <math>\mu_x\circ X_x^{-1}</math> converges [[weak convergence|weakly]] to a [[measure (mathematics)|measure]] in <math>\mathbb{D}[0,1]</math> as <math>x\to \infty</math>.<ref name="BM" />

=== Result of Bareikis-Manstavičius ===
Bareikis and Manstavičius generalized the theorem of Deshouillers-Dress-Tenenbaum and derived a limit theorem for the sum
:<math>S_x(t):=\frac{1}{x}\sum\limits_{m\leq x}\frac{M(m,m^t)}{M(m)}</math>
for a class of multiplicative functions <math>f</math> that satisfy certain analytical properties. The resulting distribution is the more general [[beta distribution]].<ref name="BM" />
== References ==
{{reflist}}

[[Category:Theorems in analytic number theory]]