Thu, 06 Jan 2005 14:48:51 +0000 Gary McGuire <gmg@maths.nuim.ie> I would naively have thought that the first digit in the primes was uniformly distributed, because I can't think of a reason why there are more primes beginning with a 1 than with a 9, but this is not true. (Anyone got a heuristic argument for this?)

From the point of view of an amateur (me) it seems intuitive that any series that grows, on average, faster than an arithmetic progression will have more elements with first digit 1 than first digit 2, and more with 2 than with 3, and so on. Hmmm, maybe not *any* series, but certainly those with increasing gap between elements, that doesn't skip digits--- so say any series that grows faster than O(n) but slower than O(1.11^n). Or more generally, for first digits in base B, slower than O((B/(B-1))^n). Yes, I'm sure the distribution of first digits isn't rigorously defined for infinite sequences, but you asked for an heuristic argument.

The basic intuition (if it isn't obvious) comes from looking at the region of the sequence between 10^i and 10^{i+1} for any given i. For any i, if sequence is more dense in [1 X 10^i, 2 X 10^i) than in [2 X 10^i, 3 X 10^i), and so on, then we'll have more leading 1's. This simple observation isn't enough for sequences that skip around between 1st digits, though.