numbers characterizing the Riesz mean. One must take . The sum over is the sum over the zeroes of the Riemann zeta function, and can be shown to be a convergent series for . Approximation by Riemann zeta zeros There is an explicit formula for the summatory Mangoldt function given by If we separate out the trivial zeros of the zeta function, which are the negative even integers, we obtain (The sum is not absolutely convergent, so we take the zeros in order of the absolute value of their imaginary part.) Taking the derivative of both sides, ignoring convergence issues, we get an "equality" of distributions Therefore, we should expect that the sum over nontrivial zeta zeros 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. Generalized von Mangoldt function The functions where denotes the Möbius function and denotes a positive integer, generalize the von Mangoldt function. The function is the ordinary von Mangoldt function . See also Prime-counting function References External links Allan Gut, Some remarks on the Riemann zeta distribution (2005) Heike, How plot Riemann zeta zero spectrum in Mathematica? (2012) Arithmetic functions