L'ORDRE DE BENFORD, UN PAS VERS UN ORDRE CACHE DANS LES NOMBRES PREMIERS

BENFORD'S ORDER, A STEP TOWARDS A HIDDEN ORDER FOR PRIMES - One might intuitively assume that the value of the first significant digit of a number chosen at random from a sequence of numerical data (such as supermarket product prices, river lengths, or distances between galaxies) is evenly distributed between 1 and 9. However, this is a cognitive bias (more precisely, an equiprobability bias) because Newcomb-Benford's law, or the first digit law, or more commonly known as Benford's law, reveals a completely different and largely uneven distribution. This law describes in particular a strictly decreasing order (the reverse natural order of the digits from 1 to 9) with regard to the probability of occurrence of this first digit; we will call it Benford's order, distinguishing it clearly from the eponymous law.As early as 1881, American astronomer S. Newcomb ([1]) presented his empirical observations on a list of numbers such as those found on the used pages of a common logarithm table: for the probabilities of occurrence of the first significant digit, he obtained a descending order from 1 to 9; he also gave those of the second digit in descending order from 0 to 9. Confirmed by American engineer F. Benford ([2]) in 1938, the occurrence of the first significant digit in a statistical distribution of numbers approximately follows a logarithmic law and decreases from 1 as the digit increases. The quantitative aspect of this law ([1], [2], [8], [9]) stipulates that about 30% of numbers begin with a 1, while less than 5% begin with a 9.Valid for many sequences of numbers (any statistical data, such as stock prices, exchange rates, but also for sequences of factorials or Fibonacci numbers), this phenomenological law also serves as a detector of tax, electoral, or other fraud. However, it does not hold true for sequences of integers or primes. Since Newcomb and Benford's findings, several authors have considered the problem of the initial digit in the case of primes ([3], [4], [5], [6], [7]), observing that if the probability that a random integer chosen from a list of numerical data has a certain c as its first digit is expressed, in base b, by the logarithmic density logb(1 + 1/c), which translates in base 10 by Benford's descending order from 1 to 9, the same is not true for the natural density of primes. We propose to continue this reflection with the help of systematic numerical experiments for various families of primes, defining them by their initial digit, by the absence of a certain digit, by the mandatory presence of a certain digit, by their second digit, by their third digit, or by their terminal digit. Our aim will be to determine whether there are families of primes that satisfy Benford's order for a certain characteristic digit of the family, and what order, if any, is satisfied by other families.If the natural density of primes is not expressed in the logarithmic form of Benford's law, we will see that, as soon as we place ourselves in an interval I(r) = [2, M = 10r] with r sufficiently large but finite, several subsets of primes satisfy Benford's order (or even the complete Benford's order from 0 to 9) with respect to a certain characteristic digit c, then tend to equiprobability for all values of c as r approaches infinity.Finally, by ranking the primes according to the m digits (m > 0) that begin their decimal expression, i.e., by their initial integer, the results obtained will make it possible to define a hidden order among the primes that generalizes Benford's order.The numerical results presented were obtained primarily using PARI/GP software, which is particularly well suited to our calculations.-