Von Mangoldt function

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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).
View a complete list of arithmetic functions

Definition

Hands-on definition

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:

where is the logarithm, is the all ones function, and denotes Dirichlet product.

By the Mobius inversion formula, this is equivalent to defining:

,

where is the Mobius function.