In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, the unit function is a completely
multiplicative function
In number theory, a multiplicative function is an arithmetic function f of a positive integer n with the property that f(1)=1 and
f(ab) = f(a)f(b) whenever a and b are coprime.
An arithmetic function is said to be completely multiplicative (o ...
on the positive integers defined as:
:
It is called the unit function because it is the
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
for
Dirichlet convolution
In mathematics, Dirichlet convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb ...
.
[.]
It may be described as the "
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
of 1" within the set of positive integers. It is also written as
(not to be confused with
, which generally denotes the
Möbius function
The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and m ...
).
See also
*
Möbius inversion formula
In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.
A large genera ...
*
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
*
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
References
Multiplicative functions
1 (number)
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