In statistics, regression toward the mean (also called reversion to the mean, and reversion to mediocrity) is the phenomenon where if one sample of a random variable is extreme, the next sampling of the same random variable is likely to be closer to its mean. Furthermore, when many random variables are sampled and the most extreme results are intentionally picked out, it refers to the fact that (in many cases) a second sampling of these picked-out variables will result in “less extreme” results, closer to the initial mean of all of the variables.

Mathematically, the strength of this “regression” effect is dependent on whether or not all of the random variables are drawn from the same distribution, or if there are genuine differences in the underlying distributions for each random variable. In the first case, the “regression” effect is statistically likely to occur, but in the second case, it may occur less strongly or not at all.

Regression toward the mean is thus a useful concept to consider when designing any scientific experiment, data analysis, or test, which intentionally selects the “most extreme” events – it indicates that follow-up checks may be useful in order to avoid jumping to false conclusions about these events; they may be “genuine” extreme events, a completely meaningless selection due to statistical noise, or a mix of the two cases.^[4

https://en.wikipedia.org/wiki/Regression_toward_the_mean

In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).

The first such distribution found is π(N) ~ N/log(N), where π(N) is the prime-counting function (the number of primes less than or equal to N) and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N). Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(10¹⁰⁰⁰) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(10²⁰⁰⁰) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N).

https://en.wikipedia.org/wiki/Prime_number_theorem

https://en.wikipedia.org/wiki/Riemann_hypothesis#:~:text=In%20mathematics%2C%20the%20Riemann%20hypothesis,unsolved%20problem%20in%20pure%20mathematics.