Numbers of primes
The fact that curve (c) must cross the axis was proved by John Littlewood in 1914, and it is known to have at least one crossing below 10317. Somewhat related to the curves shown here is the function MoebiusMu[n], equal to 0 if n has a repeated prime factor and otherwise (-1)^Length[FactorInteger[n]]. The quantity FoldList[Plus, 0, Table[MoebiusMu[i], {i, n}]] behaves very much like a random walk. The so-called Mertens Conjecture from 1897 stated that the magnitude of this quantity is less than √n. But this was disproved in 1983, although the necessary n is not known explicitly.