Definition: Von Mangoldt Function

The von Mangoldt \(\Lambda(n)\) function $\Lambda:\mathbb N\to\mathbb N$ is an arithmetic function having the value of the natural logarithm for all $n\in\mathbb N$ which are positive powers of prime numbers. \[\Lambda(n):=\begin{cases} \log(p)&\text{for } n=p^e,\; e\ge 1\\ 0&\text{for all other }n > 0 \end{cases}.\]

Example

As an example, let us calculate the von Mangoldt function for the value $n=12.$

$$\begin{array}{rcl} \Lambda(12)&=&\cancel{\Lambda(1)}+\Lambda(2)+\Lambda(3)+\Lambda(4)+\Lambda(5)+\cancel{\Lambda(6)}+\Lambda(7)+\Lambda(8)+\Lambda(9)+\cancel{\Lambda(10)}+\Lambda(11)+\cancel{\Lambda(12)}\\ &=&\Lambda(2)+\Lambda(3)+\Lambda(4)+\Lambda(5)+\Lambda(7)+\Lambda(8)+\Lambda(9)+\Lambda(11)\\ &=&\log(2)+\log(3)+\log(2)+\log(5)+\log(7)+\log(2)+\log(3)+\log(11)\\ &=&\log(2\cdot 3\cdot 2\cdot 5\cdot 7\cdot 2\cdot 3\cdot 11)\\ &=&\log(27720)\\ &\approx&10.23\\ \end{array}$$

Examples: 1
Proofs: 2 3
Propositions: 4


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References

Bibliography

  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927