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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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 \( \Lambda (n) \) 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, \( \Lambda (n) \) 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 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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