Von Mangoldt function

The von Mangoldt function satisfies the identity[1][2] The sum is taken over all integers d that divide n. This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to 0.

For example, one has The logarithmic derivative is then[6] These are special cases of a more general relation on Dirichlet series.

Von Mangoldt provided a rigorous proof of an explicit formula for ψ(x) involving a sum over the non-trivial zeros of the Riemann zeta function.

The Mellin transform of the Chebyshev function can be found by applying Perron's formula: which holds for Re(s) > 1.

Assuming the Riemann hypothesis, they demonstrate that In particular this function is oscillatory with diverging oscillations: there exists a value K > 0 such that both inequalities hold infinitely often in any neighbourhood of 0.

The graphic to the right indicates that this behaviour is not at first numerically obvious: the oscillations are not clearly seen until the series is summed in excess of 100 million terms, and are only readily visible when y < 10−5.

given by[9] 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.)

In the opposite direction, in 1911 E. Landau proved that for any fixed t > 1[10] (We use the notation ρ = β + iγ for the non-trivial zeros of the zeta function.)

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.