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.
