Mathématiques | Informatique
Luca Charlier, 2004 | La Tour-de-Trême, FR
In this work, we are interested in the asymptotic density of several subsets of the natural numbers, often relating them to sequences and sums of prime numbers. We start with the definitions of natural and Dirichlet asymptotic densities. We give two proofs that the density of primes amongst N is zero, and next prove the Euler product formula in order to analyze the density of square-free numbers which is equal to 1/Zeta(2), where Zeta is the Riemann zeta function. We then show that the density of the k-th powers and the set of all perfect powers is zero. Furthermore, we continue by proving that the new sum 1/p_1 + 1/p_2(1-1/p_1) + 1/p_3(1-1/p_1)(1-1/p_2) + … where p_i is the i-th prime, converges to 1, using the densities for which a given prime number is the smallest prime divisor, intuitively obtained from Eratosthenes› sieve. Finally, we study the Dirichlet density of numbers with an even number of digits and conjecture it is equal to 1/2.
Introduction
We are interested in computing natural and Dirichlet densities of several subsets of the natural numbers. We choose the set of primes, for its classiness; the square-free numbers, a famous case in the study of densities; the perfect powers for their intuitive access. The numbers with a given factor were chosen after the result was found by considering Eratosthenes› sieve, and finally the even-digited numbers as a sort of challenge to end the work.
Methods
We first define the natural density for its intuitive introduction to the field, but quickly continue with the Dirichlet density, which is easier to use. Some basic properties of the latter density are then given and proven, involving induction and set theory. The first theorems are shown with the help of asymptotic equivalences, and both the natural and Dirichlet densities in order to demonstrate their different uses. The density of the square-free numbers is then analyzed with the Dirichlet density, and the problem is then simplified with the Euler product theorem, which allows transforming a sum of multiplicative functions into a product of sums of fractions involving the function and prime numbers. Simplifications then lead to the desired result. We also define an explicit sequence for the series, proven through induction, and a generalization is made. The density of k-powers is analyzed in a similar fashion. Furthermore, the density of all perfect powers is estimated with an upper bound converging towards 0, which proves that the density is null.The core of the work is spread amongst a few pages, involving some identities and set theory. In the end, induction brings us to the construction of a sum of prime fractions, and its limit is computed using the divergence of the zeta function at 1. At last, we make some numerical computations on the density of integers with an even number of digits, and are able to at least prove its convergence with an estimation and basic properties of limits.
Results
We found following results : prime numbers have a 0-density, whilst square-free have density 1/Zeta(2). More generally it is shown that the density of k-free numbers is 1/Zeta(k). The density of perfect powers is 0, as for the density of all perfect powers. The density of the sets for which a given prime number p_i is the smallest prime divisor is 1/p_i(1-1/p_1)(1-1/p_2)…(1-1/p_(i-1)). The sum of those densities is equal to 1, and we show that the probability of finding such numbers at random is equal to their density. We sadly were unable to compute the Dirichlet density of the integers with an even number of digits, which is conjectured to be equal to one half. Its existence is at least shown, whereas the natural density is shown to not exist.
Discussion
The first version of the work had to be corrected in some places because one result was incorrectly applied, since density does not form a measure in general. Those issues were corrected accordingly, without much effort and without any of the theorems losing validity. The conjecture is intuitively thought to be correct, yet no proof has been found for it and the problem remains open, probably with an accessible answer, however technical.
Conclusions
This work contains some new results, as for example the whole section on least primes factors. Since the last part is not proven, some work still remains to be done. However, as interesting these densities might be, they do not bring very useful tools by themselves for the moment, and linking them to others parts of mathematics could be an interesting goal.
Appréciation de l’expert
Gauthier Leterrier
Dans son travail, Luca Charlier a étudié méthodiquement la notion de densité asymptotique de certains sous-ensembles des entiers naturels. Maîtrisant un certain nombre d’outils mathématiques, il a analysé différents exemples, comme l’ensemble des entiers sans facteur carré, ou encore celui des entiers ayant un nombre premier donné comme plus petit facteur. Cela a motivé l’étude de certaines sommes infinies, inspirées par le crible d’Ératosthène. La rigueur scientifique nécessaire à l’élaboration de tels résultats a permis au candidat d’aborder quelques aspects de la théorie des nombres.
Mention:
excellent
Prix spécial «Stockholm International Youth Science Seminar (SIYSS)» décerné par la Fondation Metrohm
Collège du Sud, Bulle
Enseignant: Laurent Karth
