Random Thoughts

This section contains some thoughts, more or less structured...

Real Periodicity

When I was young, there was in my home a pendulum clock near a regular clock. Both made a periodic sound, but their periods were different. From time to time, it appeared that both sounds happened exactly at the same time, but I always thought: "Can both sounds happen even more at the same time?". Quickly enough, I came to the following conclusion:
If a and b (a < b) are two real numbers, there exists a subsequence of the sequence S (defined below) which converges towards 0.
The sequence S is defined by: S(n) = Min_{k ∈ ℕ} | n × b − a × k |
Informally, we can represent this problem with two axes on which happen two periodic events A and B, with respective periods a and b. The terms in the sequences are the differences between the moment where the n-th event B happens and the closest A event (see figure below).

There are two situations:

• Either the numbers a and b verify the following property: there exists m, m' ∈ ℕ sch that a × m = b × m'
  In this case, all the terms S((i) of the sequence such that i mod m' = 0 are equal to 0 (null subsequence)
• Either the numbers a and b do not verify this property (let's say that they are "relatively prime numbers" by enlarging the concept to real numbers), in which case the value ok k minimizing the expression S(n) is:
  • Either k = ⌈ n × b / a ⌉
  • Or k = ⌊ n × b / a ⌋

We can besides note that when two numbers belong to ℚ, they are not relatively prime: if a = p / q and b = u / v, then the terms of the sequence will be null for the indexes multiple of p × v. Conversely, if one of the two numbers does not belong to ℚ, then a and b are relatively prime. Consequently, if two real numbers are obtained via a random variable with a support of non-null measure, then the probability that they are "relatively prime" is 1. In the cases of the clocks, there were thus high chances that it was the case (modulo the non atomicity of events).

Number of divisors

Let F be the function which associates to a number, the number of divisors of this number. Let H be the ordered sequence of natural numbers verifying: n ∈ H ⇔ ∀ x ∈ ℕ x < n ⇒ F(x) < F(n)
In other words, for a number n in H, there exists no smaller number with a number of divisors higher or equal to the one of n.
As such, it is an efficient frontier, represented for the first values in the figure below.
Elements of the sequence H (abscissa) and their number of divisors (ordinate)

We can observe the regularity of these elements on a logarithmic scale (x and y) :

Elements of the sequence H (abscissa) and their number of divisors (ordinate) with logarithmic scales

Asymptotically, if we consider that the k^th prime number is given by k × ln(k), one must choose the coefficients a_k in order to:

• Minimize ∏_k(k ln(k))^a_k
• Maximize the function ∏_k(a_k + 1)

What interests me here is the distribution of the powers on the prime factors. Intuitively, I feel that the terms of the product to maximize should be the most equal as possible, and thus that the a_k be of the form α ⁄ ln(k ln(k)), but this is not really more than an intuition (any help is welcome).
For example, if we take for the a_k a linear decrease, making the terms U(n) = p(0)^k×p(1)^k-1× ... ×p(n)¹, we realize quickly enough that one of the terms of U is not on the efficient frontier.

The prime factors powers values are given below for the first terms of H:

Display

We can see that the distribution of the prime factors powers follows approximately a shape of 1 ⁄ ln(x), letting this hypothesis open. Besides, we can once more observe that the number of terms of H is almost the same for each number size.