Deshouillers–Dress–Tenenbaum theorem

<h2 id="deshouillersdresstenenbaum-theorem">Deshouillers–Dress–Tenenbaum theorem</h2>
<p>Let 
  
    
      
        n
        ≥
        1
      
    
    {\displaystyle n\geq 1}
  
 be a natural number and fix the following notation:
</p>
<ul><li>
  
    
      
        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}
  
.</li>
<li>
  
    
      
        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}
  
.</li>
<li>
  
    
      
        T
        (
        n
        )
        :=
        T
        (
        n
        ,
        n
        )
      
    
    {\displaystyle T(n):=T(n,n)}
  
</li>
<li>
  
    
      
        τ
        (
        n
        )
        :=
        
          |
        
        T
        (
        n
        ,
        n
        )
        
          |
        
      
    
    {\displaystyle \tau (n):=|T(n,n)|}
  
</li>
<li>
  
    
      
        (
        Ω
        ,
        
          
            A
          
        
        ,
        P
        )
      
    
    {\displaystyle (\Omega ,{\mathcal {A}},P)}
  
 is a <a href="/facts/Probability_space/dEcv2VrU">probability space</a>.</li></ul>
<h3>Introduction</h3>
<p>Let 
  
    
      
        d
        :
        Ω
        →
        T
        (
        n
        )
      
    
    {\displaystyle d:\Omega \to T(n)}
  
 be a <a href="/facts/Uniform_distribution_(discrete)/M52SQP5n">uniformly distributed</a> random variable on the set of divisors of 
  
    
      
        n
      
    
    {\displaystyle n}
  
 and consider the logarithmic ratio
</p>

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

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}
  
.
<p>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 <a href="/facts/Convergence_in_distribution/pWdatFlY">converge in distribution</a> when considering subsequences indexed by <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> 
  
    
      
        
          D
          
            
              p
              
                1
              
            
          
        
        ,
        
          D
          
            
              p
              
                2
              
            
          
        
        ,
        …
      
    
    {\displaystyle D_{p_{1}},D_{p_{2}},\dots }
  
 therefore one is interested in the Césaro sum.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h3>Statement</h3>
<p>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 <a href="/facts/Ces%25C3%25A0ro_mean/ttbvyscp">Cesàro mean</a> satisfies <a href="/facts/Uniform_convergence/ecz9eNWn">uniform convergence</a> to
</p>

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)}
  
.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
<h2 id="further-results">Further Results</h2>
<p>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 <a href="/facts/Multiplicative_function/4kfolbBl">multiplicative function</a> 
  
    
      
        f
        :
        
          N
        
        →
        
          
            R
          
          
            +
          
        
      
    
    {\displaystyle f:\mathbb {N} \to \mathbb {R} _{+}}
  
 and studied the stochastic behavior of
</p>

X
        (
        n
        ,
        v
        )
        :=
        
          
            
              M
              (
              n
              ,
              v
              )
            
            
              M
              (
              n
              )
            
          
        
      
    
    {\displaystyle X(n,v):={\frac {M(n,v)}{M(n)}}}
  
,
<p>where
</p>

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)}
  
.
<h3>Result of Manstavičius-Timofeev</h3>
<p>Let 
  
    
      
        
          D
        
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle \mathbb {D} [0,1]}
  
 be the <a href="/facts/Skorokhod_space/R9Jz9QLh">Skorokhod space</a> and let 
  
    
      
        
          
            B
          
        
        (
        
          D
        
        [
        0
        ,
        1
        ]
        )
      
    
    {\displaystyle {\mathcal {B}}(\mathbb {D} [0,1])}
  
 be the <a href="/facts/Borel_%25CF%2583-algebra/jvSsfpkP">Borel σ-algebra</a>. 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]}
  
.
</p><p>Manstavičius and Timofeev studied the process 
  
    
      
        
          
            (
            
              X
              
                x
              
            
            )
          
          
            x
            ≥
            m
          
        
      
    
    {\displaystyle \left(X_{x}\right)_{x\geq m}}
  
 with
</p>

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)}}}

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

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 )).}
  
<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
<p>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 <a href="/facts/Weak_convergence_of_measures/fb0I0KAL">weakly</a> to a <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a> in 
  
    
      
        
          D
        
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle \mathbb {D} [0,1]}
  
 as 
  
    
      
        x
        →
        ∞
      
    
    {\displaystyle x\to \infty }
  
.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h3>Result of Bareikis-Manstavičius</h3>
<p>Bareikis and Manstavičius generalized the theorem of Deshouillers-Dress-Tenenbaum and derived a limit theorem for the sum
</p>

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)}}}

<p>for a class of multiplicative functions 
  
    
      
        f
      
    
    {\displaystyle f}
  
 that satisfy certain analytical properties. The resulting distribution is the more general <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Deshouillers, Jean-Marc; Dress, François; Tenenbaum, Gérald (1979). "Lois de répartition des diviseurs, 1". Acta Arithmetica (in French). 34 (4): 273–283. <a href="http://eudml.org/doc/205609" target="_blank">http://eudml.org/doc/205609</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Bareikis, Gintautas; Manstavičius, Eugenijus (2007). "On the DDT theorem". Acta Arithmetica. 126 (2): 155–168. <a href="http://eudml.org/doc/278080" target="_blank">http://eudml.org/doc/278080</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Deshouillers, Jean-Marc; Dress, François; Tenenbaum, Gérald (1979). "Lois de répartition des diviseurs, 1". Acta Arithmetica (in French). 34 (4): 273–283. <a href="http://eudml.org/doc/205609" target="_blank">http://eudml.org/doc/205609</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Deshouillers, Jean-Marc; Dress, François; Tenenbaum, Gérald (1979). "Lois de répartition des diviseurs, 1". Acta Arithmetica (in French). 34 (4): 274. <a href="http://eudml.org/doc/205609" target="_blank">http://eudml.org/doc/205609</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Bareikis, Gintautas; Manstavičius, Eugenijus (2007). "On the DDT theorem". Acta Arithmetica. 126 (2): 155–168. <a href="http://eudml.org/doc/278080" target="_blank">http://eudml.org/doc/278080</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Bareikis, Gintautas; Manstavičius, Eugenijus (2007). "On the DDT theorem". Acta Arithmetica. 126 (2): 155–168. <a href="http://eudml.org/doc/278080" target="_blank">http://eudml.org/doc/278080</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Deshouillers–Dress–Tenenbaum theorem open-in-new

Deshouillers–Dress–Tenenbaum theorem