Search for a tool
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.

Results

Von Mangoldt Function -

Tag(s) : Arithmetics

Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Thanks to your feedback and relevant comments, dCode has developed the best 'Von Mangoldt Function' tool, so feel free to write! Thank you!

# 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: $$\sum_{n=2}^{\infty}{\frac{\Lambda(n)-1}{n}}=-2\gamma$$

## Source code

dCode retains ownership of the "Von Mangoldt Function" source code. Except explicit open source licence (indicated Creative Commons / free), the "Von Mangoldt Function" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Von Mangoldt Function" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Von Mangoldt Function" are not public, same for offline use on PC, tablet, iPhone or Android !
The copy-paste of the page "Von Mangoldt Function" or any of its results, is allowed as long as you cite the online source https://www.dcode.fr/mangoldt-lambda
Reminder : dCode is free to use.

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!