Definition
The von Mangoldt function, conventionally written as Λ(n), is defined as
which is a sequence starting:
oeis sequence A014963
It is an example of an important arithmetic function that is neither multiplicative nor additive.
The von Mangoldt function satisfies the identity
that is, the sum is taken over all integers d which 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 instance, let n=12. Recall the prime factorization of 12, 12=22·3, which will turn up in the example.
- Take the summation over all distinct positive divisors d of n:
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- This provides an example of how the summation of the von Mangoldt function equals log (n).
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.
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