Tool to calculate von Mangoldt Lambda Λ function values. Mangoldt's Λ function is an arithmetic function with properties related to prime numbers.
Von Mangoldt Function - dCode
Tag(s) : Arithmetics
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Von Mangoldt Function
Lambda Λ(n) Calculator
Answers to Questions (FAQ)
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:
| n | Λ(n) |
|---|---|
| 1 | 0 |
| 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:
| n | Λ(n) |
|---|---|
| 1 | 0 |
| 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
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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Von Mangoldt Function on dCode.fr [online website], retrieved on 2024-12-21,
