Von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition. The von Mangoldt function, denoted by , is defined as The values of for the first nine positive integers (i.e. natural numbers) are which is related to . The summatory von Mangoldt function, , also known as the second Chebyshev function, is defined as Von Mangoldt provided a rigorous proof of an explicit formula for involving a sum over the non-trivial zeros of the Riemann zeta  function. This was an important part of the first proof of the prime number theorem. Properties. The von Mangoldt function satisfies the identity The sum is taken over all integers that divide . This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to . For example, consider the case . Then By Möbius inversion, we have For all formula_7, we have Also, there exist positive constants and such that for all formula_7, and for all sufficiently large . Dirichlet series. The von Mangoldt function plays an important role in the theory of Dirichlet  series, and in particular, the Riemann zeta function. For example, one has The logarithmic derivative is then These are special cases of a more general relation on Dirichlet series. If one has for a completely multiplicative function , and the series converges for , then converges for . Chebyshev function. The second Chebyshev function "ψ"("x") is the summatory function of the von Mangoldt function: The Mellin transform of the Chebyshev function can be found by applying Perron's formula: which holds for . Exponential series. Hardy and Littlewood examined the series in the limit . Assuming the Riemann hypothesis, they demonstrate that In particular this  function is oscillatory with diverging oscillations: there exists a value 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 . Riesz mean. The Riesz mean of the von Mangoldt function is given by Here, and are 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 formula_23 given by 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 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.