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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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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 \( \Lambda (n) \) (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 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 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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