von Mangoldt function

Definition

The von Mangoldt function $\Lambda$ is the arithmetic function given, for each positive integer $n$ , by $\Lambda(n) = \begin{cases} \log p & \text{if } n = p^k \text{ for a prime } p \text{ and an integer } k \ge 1, \\ 0 & \text{otherwise}. \end{cases}$ Here $\log$ is the natural logarithm. The word prime has the meaning of the definition of a prime number.

Remarks

For example, $\Lambda(8) = \log 2$ , $\Lambda(7) = \log 7$ , and $\Lambda(1) = \Lambda(6) = 0$ . For each positive integer $n$ , $\sum_{d \mid n} \Lambda(d) = \log n$ . The function $\Lambda$ is not multiplicative. It is named after Hans von Mangoldt. The sum $\psi(x) = \sum_{n \le x} \Lambda(n)$ is the second Chebyshev function. The prime number theorem is equivalent to $\psi(x) \sim x$ .

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Divisibility definition
Prime number definition
von Mangoldt function definition

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Wiener–Ikehara theorem

Tag	`087EMZTHWN`
URI	`m2://087EMZTHWN`
Kind	definition
Writing	W1 stated
MSC	11A25
Defines	von Mangoldt function
Uses	Prime number

von Mangoldt function

Definition

Remarks

Reading path

Where this leads

Further reading

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