We encountered this graph earlier. Recall that it shows us the deviations of the prime counting function π(x) from the smooth approximating function R(x). It was hinted that this noisy function might somehow decompose into fairly simple component functions. Indeed, this is the case. The usual process of Fourier analysis essentially decomposes "signals" such as this into (periodic) sine wave functions. In this case, the component functions are quasi-periodic, based on sine waves but with a particular kind of logarithmic deformation. Remarkably, the functions in question, the sum of which produces the function seen above, are intimately connected with the nontrivial zeros of the zeta function which we've just seen. << previous      beginning      next >>