The case of Mertens’ conjecture is interesting in that surface level heuristic considerations suggest that it’s not true. In particular, it violates the heuristic that I described here.
Roughly speaking, the function “the difference between the number of natural numbers up to k with an odd number of prime factors and the number of natural numbers up to k with an even number of prime factors” is supposed to be normally distributed with standard deviation ‘square root of k,’ and Merten’s conjecture predicts a truncated normal distribution.
Yes, this is right.
The case of Mertens’ conjecture is interesting in that surface level heuristic considerations suggest that it’s not true. In particular, it violates the heuristic that I described here.
Roughly speaking, the function “the difference between the number of natural numbers up to k with an odd number of prime factors and the number of natural numbers up to k with an even number of prime factors” is supposed to be normally distributed with standard deviation ‘square root of k,’ and Merten’s conjecture predicts a truncated normal distribution.