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

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*!

Tool to calculate von Mangoldt Lambda Λ function values. Mangoldt's Λ function is an arithmetic function with properties related to prime numbers.

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 $ |

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:__ $$ \begin{align}\sum_{d \mid 8} \Lambda(d) &= \Lambda(1) + \Lambda(2) + \Lambda(4) + \Lambda(8) \\ &= \Lambda(1) + \Lambda(2) + \Lambda (2^2) + \Lambda(2^3) \\ &= 0 + \ln(2) + \ln(2) + \ln(2) \\ &=\ln (2 \times 2 \times 2) \\ &= \ln(8) \end{align} $$

The Hans Von **Mangoldt** Lambda function can be used to calculate $ \gamma $ the Euler-Mascheroni constant with the la formula : $$ \sum_{n=2}^{\infty}{\frac{\Lambda(n)-1}{n}}=-2\gamma $$

dCode retains ownership of the source code of the script Von Mangoldt Function online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Von Mangoldt Function script for offline use on PC, iPhone or Android, ask for price quote on contact page !

mangoldt,lambda,function,hans,von

Source : https://www.dcode.fr/mangoldt-lambda

© 2020 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback

▲