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. The von Mangoldt function, denoted by Λ(n), is defined as The values of Λ(n) for the first nine positive integers (i.e. natural numbers) are which is related to . The von Mangoldt function satisfies the identity 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, consider the case n = 12 = 22 × 3. Then By Möbius inversion, we have and using the product rule for the logarithm we get For all , we have Also, there exist positive constants c1 and c2 such that for all , and for all sufficiently large x. 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 f (n), and the series converges for Re(s) > σ0, then converges for Re(s) > σ0. The second Chebyshev function ψ(x) is the summatory function of the von Mangoldt function: It was introduced by Pafnuty Chebyshev who used it to show that the true order of the prime counting function is . 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. This was an important part of the first proof of the prime number theorem. The Mellin transform of the Chebyshev function can be found by applying Perron's formula: which holds for Re(s) > 1. Hardy and Littlewood examined the series in the limit y → 0+. 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.