Von Mangoldt function
This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
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The von Mangoldt function, denoted , is an arithmetic function defined as follows:
- , for a prime and a positive integer.
- if has more than one prime divisor.
Definition in terms of Dirichlet product
The von Mangoldt function is the unique function such that:
By the Mobius inversion formula, this is equivalent to defining:
where is the Mobius function.