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 (sequence in the OEIS).
The summatory von Mangoldt function, ψ(x), also known as the Chebyshev function, is defined as
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 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
The von Mangoldt function plays an important role in the theory of Dirichlet series, and in particular, the Riemann zeta function. In particular, one has
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