The difference function R(x) – π(x) seen earlier can be expressed as the infinite sum over the set of zeros (both trivial and nontrivial) ρ of the Riemann zeta function:

$R(x)-\pi(x)=\sum_{\rho}R(x^{\rho}$

This sum separates into sums over the trivial and nontrivial zeros respectively. The former is the relatively simple function
R(x –2) + R(x –4) + R(x –6) + . . .

The sum over the nontrivial zeros can be expressed as the sum of the sequence of functions {–Tk(x)} where Tk is defined as follows:

$T_{k}(x)=-R(x^{\rho_{x}})-(R(x^{\rho_{-x}})$

where the ρk and ρ–k are the kth pair of nontrivial zeta zeros, which we know must be complex conjugates. The first four functions T1(x), T2(x),T3(x), and T4(x) are pictured above.

Our first apparent obstacle is that ρk and ρ–k are complex numbers. However, the function xk can be meaningfully extended from real k to complex k in a fairly straightforward way. This means that the x ρk and x –ρk are also complex-valued.

This also initially seems like a problem, as the Riemann function R defined earlier as an approximation of π(x) was clearly intended to act on real values only. However, by the same process of analytic continuation discussed earlier, R can be extended to the entire complex plane, taking the form given by the Gram series:

$R(x)=1+\sum_{k=1}^{\infty}\frac{(\ln x)^{k}}{kk!\zeta(k+1)}$

Here ln x is the usual extension of the logarithm function to . Also note the role of the Riemann zeta function.

So we see that R(x ρk) and R(x ρ–k) can be given precise meanings, and will yield complex numbers. Usefully, the imaginary parts of this pair of complex numbers can be shown to cancel, so that their sum which is Tk(x) will always be a real-valued function.

A detailed explanation of how the nontrivial zeros can be used to exactly reconstruct the function π(x) can be found here.

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