Suppose that f(N) is a function from the natural numbers to the reals that has infinity as a limit. Let X be the set of natural numbers who eventually go below f(N). Then the logarithmic density of X is 1.
Here logarithmic density is the limit of the following ratio:
sum(1/n for n in X and < N)
/ sum(1/n for n < N)
Now you just have to pick f that grows very, very slowly. For example f(N) = log(log(log(x)))).
His actual result is this.
Suppose that f(N) is a function from the natural numbers to the reals that has infinity as a limit. Let X be the set of natural numbers who eventually go below f(N). Then the logarithmic density of X is 1.
Here logarithmic density is the limit of the following ratio:
Now you just have to pick f that grows very, very slowly. For example f(N) = log(log(log(x)))).