Luca Seth Charlier

Swiss Confederation

Schweizer Jugend forscht

On Prime Sequences and Series

Studied by Means of Asymptotic Densities

Not all natural numbers are alike, and some have properties others do not. Thus, by grouping the integers that share a common property into sets, we see that these are again not all alike, some containing more numbers than others. One way of measuring the importance, the ”size” of a subset of the natural numbers, is the asymptotic density. For example, even numbers represent half of all integers, meaning that their density is one half. Starting from an intuitive definition of density, and using several theorems dating back to Euler as well as more recent mathematicians, various sets were studied, most of them related to primes. Doing so, it was possible to compute their densities, and when possible, to relate them to sums or sequences of prime numbers. For example, as even numbers have a density of one half, and multiples of three have a density of one third, it is necessary for multiples of three which are not even (therefore multiples of three which are not multiples of six) to have a density of 1/3 – 1/6 = 1/3 (1-1/2). Continuing in that fashion and including all the integers {0, 1, -1, 2, -2, …}, we then add up the densities of all the sets for which a given prime is the smallest prime factor, which gives us 1, the density of all natural numbers :

1 = 1/2 + 1/3(1-1/2) + 1/5(1-1/3)(1-1/2) + …

Although the project started from a simple and accessible base, it led to beautiful and complex results, some of which are original.