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 formula_1 of a natural number formula_2 within the interval formula_3, where the divisor formula_1 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.
Deshouillers–Dress–Tenenbaum theorem.
Let formula_5 be a natural number and fix the following notation:
Introduction.
Let formula_15 be a uniformly distributed random variable on the set of divisors of formula_2 and consider the logarithmic ratio
formula_17,
notice that the realizations of the random variable formula_18 are characterized entirely by the divisors of formula_2 and each divisor has probability formula_20. The distribution function of formula_18 is defined as
formula_22 for formula_23.
It is easy to see that the sequence formula_24 does not converge in distribution when considering subsequences indexed by prime numbers formula_25 therefore one is interested in the Césaro sum.
Statement.
Let formula_26 be a sequence of the above-defined random variables and let formula_27. Then for all formula_28 the Cesàro mean satisfies uniform convergence to
formula_29.
Further Results.
Eugenijus Manstavičius, Gintautas Bareikis, and Nikolai Timofeev extended the theorem by replacing the counting function formula_30 in formula_31 with a multiplicative function formula_32 and studied the stochastic behavior of
formula_33,
where
formula_34.
Result of Manstavičius-Timofeev.
Let formula_35 be the Skorokhod space and let formula_36 be the Borel σ-algebra. For formula_37, define a discrete measure formula_38, describing the probability of selecting formula_39 from formula_40 with probability formula_41.
Manstavičius and Timofeev studied the process formula_42 with
formula_43
for formula_44 and the image measure formula_45 on formula_35.
That is, the image measure is defined for formula_47 as follows:
formula_48
They showed that if formula_49 for every prime number formula_50 and formula_51 for all prime numbers formula_50 and all formula_53, then formula_45 converges weakly to a measure in formula_35 as formula_56.
Result of Bareikis-Manstavičius.
Bareikis and Manstavičius generalized the theorem of Deshouillers-Dress-Tenenbaum and derived a limit theorem for the sum
formula_57
for a class of multiplicative functions formula_58 that satisfy certain analytical properties. The resulting distribution is the more general beta distribution.