In computational complexity, one typically ignores multiplicative constants, but in number theory, one often doesn't. The prime number theorem is of this stron

In computational complexity, one typically ignores multiplicative constants, but in number theory, one often doesn't. The prime number theorem is of this stronger form, and it is in fact only the natural logarithm that makes the usual statement true. It is considerably more difficult to prove this asymptotic result (first proven 1896) than to get "within a multiplicative constant" as in big-O notation (first proven 1848--50).