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 Λ ( n ) {\displaystyle \Lambda (n)} , is defined as
Λ ( n ) = { log p if n = p k for some prime p and integer k ≥ 1 , 0 otherwise. {\displaystyle \Lambda (n)={\begin{cases}\log p&{\text{if }}n=p^{k}{\text{ for some prime }}p{\text{ and integer }}k\geq 1,\\0&{\text{otherwise.}}\end{cases}}}
The first few values of Λ ( n ) {\displaystyle \Lambda (n)} are
0 , log 2 , log 3 , log 2 , log 5 , 0 , log 7 , log 2 , log 3 , 0 , log 11 , 0 , … {\displaystyle 0,\log 2,\log 3,\log 2,\log 5,0,\log 7,\log 2,\log 3,0,\log 11,0,\dots }
which is related to (sequence A014963 in the OEIS).
Properties The von Mangoldt function satisfies the identity
log ( n ) = ∑ d ∣ n Λ ( d ) . {\displaystyle \log(n)=\sum _{d\mid n}\Lambda (d).}
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
∑ d ∣ 12 Λ ( d ) = Λ ( 1 ) + Λ ( 2 ) + Λ ( 3 ) + Λ ( 4 ) + Λ ( 6 ) + Λ ( 12 ) = Λ ( 1 ) + Λ ( 2 ) + Λ ( 3 ) + Λ ( 2 2 ) + Λ ( 2 × 3 ) + Λ ( 2 2 × 3 ) = 0 + log ( 2 ) + log ( 3 ) + log ( 2 ) + 0 + 0 = log ( 2 × 3 × 2 ) = log ( 12 ) . {\displaystyle {\begin{aligned}\sum _{d\mid 12}\Lambda (d)&=\Lambda (1)+\Lambda (2)+\Lambda (3)+\Lambda (4)+\Lambda (6)+\Lambda (12)\\&=\Lambda (1)+\Lambda (2)+\Lambda (3)+\Lambda \left(2^{2}\right)+\Lambda (2\times 3)+\Lambda \left(2^{2}\times 3\right)\\&=0+\log(2)+\log(3)+\log(2)+0+0\\&=\log(2\times 3\times 2)\\&=\log(12).\end{aligned}}}
By Möbius inversion, we have
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