An Erdős-Kac theorem for Smooth and Ultra-smooth integers
We prove an Erdős-Kac type of theorem for the set . If is the number of prime factors of , we prove that the distribution of for is Gaussian for a certain range of using method of moments. The advantage of the present approach is that it recovers classical results for the range where , with a much simpler proof.
For an integer , let denote the number of distinct prime divisors of . In 1940, Erdős and Kac  in their celebrated work studied the distribution of in the interval . The theorem states that for any real number , we have
where is the normal distribution function defined by
There are several proofs of Erdős-Kac Theorem. For instance, it has been proved by Billingsley  and Granville and Soundararajan  using the method of moments and sieve theory. Different variations of this theorem have been considered by several authors. In the present note, we shall study the Erdős-Kac theorem for smooth numbers. Recall that
is the set of smooth integers, where is defined as the largest prime factor of , with the convention . Also, recall that we set
The main goal of this result is to prove an analogue of (1) with the set in the range
where, as always,
Hildebrand , Alladi , and Hensley  have considered the distribution of prime divisors of smooth integers in different ranges of .
Hensley proved an Erdős-Kac type theorem when lies in the range
By using different method Alladi obtained an analogue of the Erdős-Kac Theorem for the following range
Later, Hildebrand extended previous results to include the range
which is a completion of Alladi and Hensley’s results.
Although (2) does not cover Alladi’s, Hensley’s and Hildebrand’s ranges, our applied method is completely different and much easier than the methods used by previous authors.
Our approach is based on the method of moments as Billinglsley used in . We will introduce some approximately independent random variables, and by the Central Limit Theorem, we shall show that this random variables have a normal distribution, then by applying method of moments we get our desired result in (1).
The first step of the proof is to apply a truncation on number prime factors. This idea is from original proof of Erdős-Kac Theorem .
For a given real number , set
then is a function that helps us to sieve out all primes exceeding , and we will show the contribution of sieved primes is negligible in understanding the distribution of . Before stating the main result, we begin introducing some notation. Let is the number of distinct prime divisors of a smooth number, namely
where is and according to the prime divides or not.
Let be the mean value of , more formally
and is the variance of , defined by
Now we are ready to state the main theorem.
For any real number , we have
holds in the range (2).
Theorem 1.1 is proved in Section . The proof relies on the method of moments and the estimates for .
be the set of ultra-smooth integers whose canonical decomposition is free of prime powers exceeding , where
We also have the following theorem
For any real number , we have
holds in the range (2).
Considering this relation between the local behaviour of and gives us a similar proof as Theorem 1.1, so we shall avoid proving this theorem.
I would like to thank Andrew Granville and Dimitris Koukoulopoulos for all their advice and encouragement as well as their valuable comments on the earlier version of the present paper. I am also grateful to Adam Harper, Sary Drappeau and Oleksiy Klurman for helpful conversations.
Here we briefly recall some standard facts from probability theory (See Feller  for more details) and we shall give a few important lemmas.
If a random variable converges to in probability, particularly , then a second random variable (on the same probability space) tend to in distribution if and only if in distribution.
If distribution function satisfying as , for , then for each .
If for each x, and if is bounded in for some positive , then, .
(A special case of the central limit theorem): If are independent and uniformly bounded random variables with mean and finite variance and if diverges then the distribution of converges to the normal distribution function.
By recalling [4, Theorem 2.4.] for , and , we have
where and denotes the saddle point of the Perron’s integral for , which is the solution of the following equation
This function will play an important role in this work, so we briefly recall some fundamental facts about this function. By [4, Lemma3.1], for any , we have the following estimate for
where is a unique real non-zero root of the equation
and when , we have
By [4, Lemma 4.1], we have the following important estimate
(De la Breteche, Tenenbaum) For any , uniformly we have
Here we use a particular case of Lemma 2.1. If the range of is restricted to , we get
For , we define
By using the saddle point method, Tenenbaum and de la Breteche in  obtained an estimate for the expectation and the variance of . First, we define
We state the following lemma from .
(Tenenbaum, de la Breteche) we have uniformly for
We now study the expectation of , where .
If , then we have
and the proof is complete. ∎
If and , then we have
since is bounded. By the given estimate for in (6) and using Mertens’ estimate, we obtain
By applying the estimate of in (7), we get our desired result. ∎
Here we will introduce a truncated version of and in the following lemma and corollary we show that the contribution of large prime factors does not affect the expected value of number of prime factors of and hence the distribution of , when is small enough. We define
If , then we have
By Lemma 2.4, we have
and we have our desired result. ∎
Now we define
In the following lemma we will show can be replaced by in the statement of Theorem 1.1.
Let , then we have
where denotes the probability value.
3. Proof of Theorem 1.1
We begin this section by setting some random variables on a probability space and one variable for each prime , which satisfies
The random variables ’s are independent.
Now we define the partial sum as follows
By the definition of ’s and the estimate in (5) and (9), we deduce that has a mean value and variance of the order in the range , this means that and have roughly the same variance and the same mean value.
In the following lemma we get an upper bound for the difference of moments of and , where
If , then for any positive integer , we have
By the definition of and , we have
So for the difference of moment, we have
Without loss of generality we assume that ’s are distinct, then by using the estimate (5), we have
The main terms in the above subtraction are the same and will be eliminated. Therefore,
If , then . So we can ignore the term . Thus,
We now use Lemma 2.5, and we get the following upper bound for each
Proof of Theorem 1.1.
We start our proof by normalizing the random variable . Define
By recalling the central limit theorem, one can say that has a normal distribution , since ’s are independent. We set
By using the method of moments, we will show that the moments of are very close to those corresponding sum and they both converge to the moment of normal distribution for every positive integer .
By the multinomial theorem, we have
Now using Lemma 2.3, we have
We showed that the difference of moments goes to for large values of . By the remark (2), we conclude that two random variables and have a same distribution.
By Remark 4, the random variable has a normal distribution. It remains to show that the moments of are very close to those of the normal distribution.
By recalling Remark 3, we need to prove that the moment are bounded in when increases.
In fact, we will show that for each
To complete the proof, we define the random variables , which are independent.
Where is over -tuple , where are positive integers, and .
By the definition of , we have .
To avoid zero terms, we can assume that for each . Also we have . Thus,
Therefore, the value of inner sum in (27) is at most
Each is strictly greater than , and we have , therefore and this implies that
-  Krishnaswami Alladi. An Erdős-Kac theorem for integers without large prime factors. Acta Arith., 49(1):81–105, 1987.
-  Patrick Billingsley. On the central limit theorem for the prime divisor functions. Amer. Math. Monthly, 76:132–139, 1969.
-  R. de la Bretèche and G. Tenenbaum. Entiers friables: inégalité de Turán-Kubilius et applications. Invent. Math., 159(3):531–588, 2005.
-  Régis de la Bretèche and Gérald Tenenbaum. Propriétés statistiques des entiers friables. Ramanujan J., 9(1-2):139–202, 2005.
-  P. Erdös and M. Kac. The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math., 62:738–742, 1940.
-  William Feller. An introduction to probability theory and its applications, volume 1. The name of the publisher, 3 edition, 1968.
-  Andrew Granville and K. Soundararajan. Sieving and the Erdős-Kac theorem. In Equidistribution in number theory, an introduction, volume 237 of NATO Sci. Ser. II Math. Phys. Chem., pages 15–27. Springer, Dordrecht, 2007.
-  Douglas Hensley. The distribution of among numbers with no large prime factors. In Analytic number theory and Diophantine problems (Stillwater, OK, 1984), volume 70 of Progr. Math., pages 247–281. Birkhäuser Boston, Boston, MA, 1987.
-  Adolf Hildebrand. On the number of prime factors of integers without large prime divisors. J. Number Theory, 25(1):81–106, 1987.
-  Gérald Tenenbaum. On ultrafriable integers. Q. J. Math., 66(1):333–351, 2015.