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Von Mangoldt Function

Tool to calculate von Mangoldt Lambda Λ function values. Mangoldt's Λ function is an arithmetic function with properties related to prime numbers.

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Von Mangoldt Function -

Tag(s) : Arithmetics

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# Von Mangoldt Function

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## Lambda Λ(n) Calculator

Tool to calculate von Mangoldt Lambda Λ function values. Mangoldt's Λ function is an arithmetic function with properties related to prime numbers.

## Answers to Questions

### What is the Von Mangoldt Lambda function? (Definition)

The function $$\Lambda (n)$$ is defined by: $$\Lambda (n)= {\begin{cases}\ln p & {\mbox{if }}n=p^{k} \\ 0 & {\mbox{else}} \end{cases} }$$

with $$p$$ a prime number and $$k \in \mathbb{N}, k \geq 1$$ (a nonzero positive integer).

This is the natural logarithm $$\log (n) = \ln (n)$$

Example: The values of $$\Lambda (n)$$ for the first values of $$n$$ are:

nΛ(n)
10
2$$\ln 2$$
3$$\ln 3$$
4$$\ln 2$$
5$$\ln 5$$
6$$0$$
7$$\ln 7$$
8$$\ln 2$$
9$$\ln 3$$

### What are the properties of the Von Mangoldt Lambda function?

By its definition, $$\Lambda (n)$$ allows to describe the value of the natural logarithm $$\ln n$$ : $$\ln n=\sum _{d\mid n}\Lambda (d)$$ with $$d$$ a natural integer that divides $$n$$.

Example: \begin{align}\sum_{d \mid 8} \Lambda(d) &= \Lambda(1) + \Lambda(2) + \Lambda(4) + \Lambda(8) \\ &= \Lambda(1) + \Lambda(2) + \Lambda (2^2) + \Lambda(2^3) \\ &= 0 + \ln(2) + \ln(2) + \ln(2) \\ &=\ln (2 \times 2 \times 2) \\ &= \ln(8) \end{align}

### What is the link with the Euler–Mascheroni gamma constant?

The Von Mangoldt Lambda function can be used to calculate $$\gamma$$ the Euler-Mascheroni constant with the la formula : $$\sum_{n=2}^{\infty}{\frac{\Lambda(n)-1}{n}}=-2\gamma$$

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## Questions / Comments

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