Someone asked on Math Overflow about the distribution of digits in primes. It seems 0 is the least common digit and 1 the most common digit.

Dan Piponi replies “this is probably just a combination of general properties of sets of numbers with a density similar to the primes and the fact that primes end in 1, 3, 7 or 9” and supports this by showing that “fake primes” have very similar digit distributions as actual primes. He generates the *n*th fake prime by starting with *n* log *n* and generating a nearby random integer ending in 1, 3, 7, or 9.

It seems like this fake prime function could be useful for studying more questions. Here is Dan Piponi’s Mathematica implementation:

fakePrime[n_] := With[ {m = n Log[n]}, 10 RandomInteger[{Floor[0.09 m], Floor[0.11 m]}] + RandomChoice[{1, 3, 7, 9}] ]

Compare to the CramÃ©r random model for example here: https://terrytao.wordpress.com/2019/08/26/large-prime-gaps-and-probabilistic-models/

I don’t know if there are other random “models” of the primes.