Von Mangoldt Function

The von Mangoldt function, denoted by the Greek letter capital lambda and named after the German mathematician Hans von Mangoldt, is an arithmetic function in number theory that is neither multiplicative nor additive. It is defined to equal the natural logarithm of a prime p when its input n is a prime power p to the k for some k of at least 1, and to equal zero for every other positive integer, so its values begin 0, log 2, log 3, log 2, log 5, 0, log 7, log 2, log 3, 0, log 11, 0 and so on. A basic identity states that summing the von Mangoldt function over every divisor of n gives the natural logarithm of n, a consequence of the fundamental theorem of arithmetic. The functions summatory version, the Chebyshev function, was used by Pafnuty Chebyshev to show that the prime counting function has order x over log x, and von Mangoldts own rigorous explicit formulas connecting it to the non-trivial zeros of the Riemann zeta function contributed to the first proof of the prime number theorem. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/