Deshouillers–Dress–Tenenbaum theorem

<p>The Deshouillers–Dress–Tenenbaum theorem (or in short DDT theorem) is a result from <a href="/facts/Probabilistic_number_theory/VqhyWEbD">probabilistic number theory</a>, which describes the <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> of a <a href="/facts/Divisor/DKNa2GvC">divisor</a> 
  
    
      
        d
      
    
    {\displaystyle d}
  
 of a <a href="/facts/Natural_number/u65og8JE">natural number</a> 
  
    
      
        n
      
    
    {\displaystyle n}
  
 within the interval 
  
    
      
        [
        1
        ,
        n
        ]
      
    
    {\displaystyle [1,n]}
  
, where the divisor 
  
    
      
        d
      
    
    {\displaystyle d}
  
 is chosen <a href="/facts/Uniform_distribution_(discrete)/M52SQP5n">uniformly</a>. More precisely, the theorem deals with the sum of distribution functions of the <a href="/facts/Logarithm/QeXaV97q">logarithmic</a> ratio of divisors to growing intervals. The theorem states that the <a href="/facts/Ces%25C3%25A0ro_sum/ttbvyscp">Cesàro sum</a> of the distribution functions converges to the <a href="/facts/Arcsine_distribution/KBWLLS3k">arcsine distribution</a>, 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.
</p><p>The theorem was proven in 1979 by the <a href="/facts/France/CxIeIKM3">French</a> mathematicians <a href="/facts/Jean-Marc_Deshouillers/HSJcL3l0">Jean-Marc Deshouillers</a>, François Dress, and <a href="/facts/G%25C3%25A9rald_Tenenbaum/pytj23Hm">Gérald Tenenbaum</a>. The result was generalized in 2007 by Gintautas Bareikis and Eugenijus Manstavičius.
</p>

Deshouillers–Dress–Tenenbaum theorem open-in-new

Deshouillers–Dress–Tenenbaum theorem