Von Mangoldt function
These are special cases of a more general relation on Dirichlet series. If one has
If we separate out the trivial zeros of the zeta function, which are the negative even integers, we obtain
Taking the derivative of both sides, ignoring convergence issues, we get an "equality" of distributions
peaks at primes. In fact, this is the case, as can be seen in the adjoining graph, and can also be verified through numerical computation.
The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to the imaginary parts of the Riemann zeta function zeros. This is sometimes called a duality.