In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, functions of
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of
coprime numbers, and such functions are called
multiplicative functions. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article.
Definition
A completely multiplicative function (or totally multiplicative function) is an
arithmetic function (that is, a function whose
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
is the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s), such that ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'')''f''(''b'') holds ''for all'' positive integers ''a'' and ''b''.
Without the requirement that ''f''(1) = 1, one could still have ''f''(1) = 0, but then ''f''(''a'') = 0 for all positive integers ''a'', so this is not a very strong restriction.
The definition above can be rephrased using the language of algebra: A completely multiplicative function is a
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
from the
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
(that is, the positive integers under multiplication) to some other monoid.
Examples
The easiest example of a completely multiplicative function is a
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
with leading coefficient 1: For any particular positive integer ''n'', define ''f''(''a'') = ''a''
''n''. Then ''f''(''bc'') = (''bc'')
''n'' = ''b''
''n''''c''
''n'' = ''f''(''b'')''f''(''c''), and ''f''(1) = 1
''n'' = 1.
The
Liouville function is a non-trivial example of a completely multiplicative function as are
Dirichlet characters, the
Jacobi symbol
Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a ...
and the
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...
.
Properties
A completely multiplicative function is completely determined by its values at the prime numbers, a consequence of the
fundamental theorem of arithmetic. Thus, if ''n'' is a product of powers of distinct primes, say ''n'' = ''p''
''a'' ''q''
''b'' ..., then ''f''(''n'') = ''f''(''p'')
''a'' ''f''(''q'')
''b'' ...
While the
Dirichlet convolution of two multiplicative functions is multiplicative, the Dirichlet convolution of two completely multiplicative functions need not be completely multiplicative.
There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function ''f'' is multiplicative then it is completely multiplicative if and only if its
Dirichlet inverse
In mathematics, the Dirichlet 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 are two arithmetic ...
is
where
is the
Möbius function.
Completely multiplicative functions also satisfy a distributive law. If ''f'' is completely multiplicative then
where ''*'' represents the
Dirichlet product
In mathematics, the Dirichlet 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 are two arithmetic ...
and
represents
pointwise multiplication.
[Apostol pg. 49] One consequence of this is that for any completely multiplicative function ''f'' one has
which can be deduced from the above by putting both
, where
is the
constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic propertie ...
.
Here
is the
divisor function.
Proof of distributive property
:
Dirichlet series
The L-function of completely (or totally)
multiplicative Dirichlet series satisfies
:
which means that the sum all over the natural numbers is equal to the product all over the prime numbers.
See also
*
Arithmetic function
*
Dirichlet L-function
*
Dirichlet series
*
Multiplicative function
References
Multiplicative functions