= Von Mangoldt function
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{title2=$\Lambda(n)$}
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The <Von Mangoldt function> assigns a <natural logarithm> weight to each positive power of a <prime number>. For a <positive integer> $n$,
$$
\Lambda(n)=\begin{cases}
\log p,&n=p^k,\quad p\text{ prime},\ k\geq1,\\
0,&\text{otherwise}.
\end{cases}
$$
Here and throughout the <Chebyshev functions>, \b[$\log$ means the <natural logarithm>]. In particular, $\Lambda(1)=0$: the exponent must be positive, even though the convention in <prime power> allows exponent zero. The <support of a function> of $\Lambda$ therefore consists precisely of the <prime powers> greater than one. <254A – analytic prime number theory/254A, Notes 1: Elementary multiplicative number theory>{ref}
For example, $\Lambda(2)=\Lambda(4)=\Lambda(8)=\log2$ and $\Lambda(3)=\Lambda(9)=\log3$, whereas $\Lambda(6)=\Lambda(12)=0$. A <composite number> can have a nonzero weight: $8$ has just one distinct <prime factor>. But $12=2^2\cdot3$ has two distinct <prime factors>, so its weight is zero. At $p^k$ the weight is \b[$\log p$, not $\log(p^k)=k\log p$]. This choice lets successive powers record successive copies of the same <prime factor> in the <Von Mangoldt divisor identity>.
The <Von Mangoldt function> is useful because it connects two views of the <prime numbers>: the <prime factorization> of an individual <integer>, through a <divisor sum>, and the accumulated weights below a cutoff, through the <Second Chebyshev function>. The latter turns questions about the distribution of <prime numbers> into questions about a <partial sum> of an <arithmetic function>.
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