As a first application, let us prove an elementary property of the von Mangoldt function $\Lambda$ and, using this property, an elementary result concerning prime numbers and factorials.

Proposition: Natural Logarithm Sum of von Mangoldt Function Over Divisors

For every $n\in\mathbb N,$ $n > 0,$ the sum of the von Mangoldt function $\Lambda$ over all divisors equals the natural logarithm of $n,$ formally

$$\sum_{d\mid n}\Lambda(d)=\log(n).\quad\quad ( * )$$

Proofs: 1

Proofs: 1

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  1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
  2. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927