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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

## Lambda Λ(n) Calculator

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

The function $\Lambda (n)$ (called Mangoldt Lambda) 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 first Lambda function values?

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$\ln 3$

It is possible to calculate the values of $\exp{\Lambda}(n)$ in order to always obtain integers, see the OEIS sequence here (link)

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

By its definition, the Von Mangoldt Lambda function $\Lambda (n)$ 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 Hans 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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