📂Number Theory

Analytic Number Theory and the Mangoldt Function

Definition ¹

The arithmetic function defined as follows $\Lambda$ is called the Mangoldt function. $$ \Lambda (n) := \begin{cases} \log p & n = p^{m} , p \text{ is prime}, m \in \mathbb{N} \\ 0 & \text{otherwise} \end{cases} $$

Basic Properties

• [1] Mangoldt series: equals the logarithmic function $\log$. In other words, $$ \sum_{d \mid n} \Lambda ( d ) = \log n $$

Explanation

$$ \begin{matrix} n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \Lambda (n) & 0 & \log 2 & \log 3 & \log 2 & \log 5 & 0 & \log 7 & \log 2 & \log 3 & 0 \\ \sum_{d \mid n} \Lambda (d) & 0 & \log 2 & \log 3 & \log 4 & \log 5 & \log 6 & \log 7 & \log 8 & \log 9 & \log 10 \end{matrix} $$ The logarithmic function is especially important in analytic number theory, as it is not only necessary for defining the derivative of arithmetic functions but also a key element in the prime number theorem.

Proof

[1]

Let’s consider primes $p_{1} , \cdots , p_{r}$ and natural numbers $a_{1} , \cdots , a_{r}$ such that $n = p_{1}^{a_{1}} \cdots p_{r}^{a_{r}}$. Then, $$ n = \prod_{k=1}^{r} p_{k}^{a_{k}} \iff \log n = \sum_{k=1}^{r} a_{k} \log p_{k} $$ according to the definition of the Mangoldt function, $$ \begin{align*} \sum_{d \mid n} \Lambda (d) =& \sum_{k=1}^{r} \sum_{m=1}^{a_{k}} \Lambda \left( p_{k}^{m} \right) \\ =& \sum_{k=1}^{r} \sum_{m=1}^{a_{k}} \log p_{k} \\ =& \sum_{k=1}^{r} a_{k} \log p_{k} \\ =& \log n \end{align*} $$

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