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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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Tag(s) : Arithmetics

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Von Mangoldt Function

Lambda Λ(n) Calculator


See also: Logarithm

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 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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