= Von Mangoldt divisor identity
{c}
{title2=$\sum_{d\mid n}\Lambda(d)=\log n$}
For every <positive integer> $n$, the <divisor sum> of its <Von Mangoldt function> weights recovers its <natural logarithm>:
$$
\sum_{d\mid n}\Lambda(d)=\log n.
$$
The sum runs over the positive <integer divisors> of $n$. For $n=1$, its only term is $\Lambda(1)=0=\log1$. Equivalently, if $L(n)=\log n$ and $\mathbf1$ is the <constant-one arithmetic function>, then $\mathbf1*\Lambda=L$, where $*$ is <Dirichlet convolution>. <254A – analytic prime number theory/254A, Notes 1: Elementary multiplicative number theory>{ref}
For $12=2^2\cdot3$, the positive <integer divisors> are $1,2,3,4,6,12$. Only $2,3,4$ contribute:
$$
\Lambda(1)+\Lambda(2)+\Lambda(3)+\Lambda(4)+\Lambda(6)+\Lambda(12)
=0+\log2+\log3+\log2+0+0
=\log12.
$$
Thus the vanishing of $\Lambda(12)$ loses no information here: the <divisor sum> retrieves both copies of the <prime factor> $2$ from the <integer divisors> $2$ and $4$.
\Image[/von-mangoldt-divisor-contributions.png]
{title=The <Von Mangoldt divisor identity> assembles $\log12$ from three nonzero divisor weights}
{id=image-von-mangoldt-divisor-contributions}
{description=The six positive <integer divisors> of $12$ appear in increasing order. Blue bars contribute $\log2$ at $2$ and $4$; the orange bar contributes $\log3$ at $3$. The remaining three weights are zero. The adjacent stacked bar shows the total $2\log2+\log3=\log12$.}
Proof outline: by the <fundamental theorem of arithmetic>, write $n=\prod_{i=1}^r p_i^{a_i}$ with distinct <prime numbers> $p_i$ and positive exponents $a_i$. Among the positive <integer divisors>, the ones with nonzero <Von Mangoldt function> weights are exactly $p_i,p_i^2,\ldots,p_i^{a_i}$. Consequently
$$
\sum_{d\mid n}\Lambda(d)=\sum_{i=1}^r\sum_{j=1}^{a_i}\log p_i
=\sum_{i=1}^r a_i\log p_i
=\log\!\left(\prod_{i=1}^r p_i^{a_i}\right)=\log n.
$$
The <natural logarithm> turns the <prime factorization> product into a sum, and the <prime powers> greater than one supply exactly the required multiplicities.
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