Logarithm In Terms of Von Mangoldt Function

Theorem

\[ \ln(n) = 1 \ast \Lambda(d) = \sum_{d \mid n} \Lambda(d)\]

where \(\Lambda\) is the von Mangoldt function and \(\ast\) is the Dirichlet convolution.

Proof

Let \(n = p_1^{\alpha_1} \dots p_k^{\alpha_k}\) as per the fundamental theorem of arithmetic. Then we have that

\[\begin{align*} \sum_{d \mid n} \Lambda(d) &= \sum_{d \mid n} \left\{\begin{matrix} \ln(p) & d = p^k \ \text{for some}\ k \geq 1 \\ 0 & \text{otherwise} \\ \end{matrix} \right\} \\ &= \sum_{i = 1}^k \sum_{j = 1}^{\alpha_i} \ln(p_i) \\ &= \sum_{i = 1}^k \alpha_i \ln(p_i) \\ &= \sum_{i = 1}^k \ln(p_i^{\alpha_i}) \\ &= \ln\left(\prod_{i = 1}^k p_i^{\alpha_i}\right) \\ &= \ln(n). \\ \end{align*}\]