If we define the sequence of functions:

$R_{n}(x)=R(x)-\sum_{m=1}^{\infty}R(x^{-2m})+\sum_{k=1}^{n}T_{k}(x)$

then from what we have seen, this should approach π(x) in limit as n tends to infinity. This is indeed what happens. Above we see R10(x) and R29(x), which clearly appear to be progressing toward a step function. An animation depicting this progression can be found here.

Here we have an infinite sequence of smooth functions, each built up from the simple Riemann function R(x) and a finite number of the quasi-sinusoidal functions Tn(x) which we have just defined. The limit of this sequence is a discontinuous function, the familiar π(x).

In this way we see how the distribution of prime numbers, as characterised by the counting function π(x), can be reconstructed from the nontrivial zeros of the Riemann zeta function in the complex plane. The function R(x) deals with the average behaviour of π(x), whereas the sum of the Tk(x) captures the local fluctuations in the distribution. For this reason the sequence of nontrivial zeta zeros is sometimes described as being "dual" to the sequence of prime numbers. They appear to be two aspects of the same thing.


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