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
of a natural number
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.
[1] The result was generalized in 2007 by Gintautas Bareikis and Eugenijus Manstavičius.
be a natural number and fix the following notation: Let
be a uniformly distributed random variable on the set of divisors of
and consider the logarithmic ratio notice that the realizations of the random variable
are characterized entirely by the divisors of
and each divisor has probability
is defined as It is easy to see that the sequence
does not converge in distribution when considering subsequences indexed by prime numbers
therefore one is interested in the Césaro sum.
be a sequence of the above-defined random variables and let
the Cesàro mean satisfies uniform convergence to Eugenijus Manstavičius, Gintautas Bareikis, and Nikolai Timofeev extended the theorem by replacing the counting function
and studied the stochastic behavior of where Let
be the Skorokhod space and let
, define a discrete measure
, describing the probability of selecting
Manstavičius and Timofeev studied the process
and the image measure
That is, the image measure is defined for
for every prime number
for all prime numbers
converges weakly to a measure in
[2] Bareikis and Manstavičius generalized the theorem of Deshouillers-Dress-Tenenbaum and derived a limit theorem for the sum for a class of multiplicative functions
that satisfy certain analytical properties.
The resulting distribution is the more general beta distribution.