# Multiplicative function

In number theory, a **multiplicative function** is an arithmetic function *f*(*n*) of the positive integer *n* with the property that *f*(1) = 1 and whenever
*a* and *b* are coprime, then

*f*(*ab*) =*f*(*a*)*f*(*b*).

An arithmetic function *f*(*n*) is said to be **completely (totally) multiplicative** if *f*(1) = 1 and *f*(*ab*) = *f*(*a*) *f*(*b*) holds *for all* positive integers *a* and *b*, even when they are not coprime.

Outside number theory, the term **multiplicative** is usually used for functions with the property *f*(*ab*) = *f*(*a*) *f*(*b*) for *all* arguments *a* and *b*; this requires either *f*(1) = 1, or *f*(*a*) = 0 for all *a* except *a* = 1. This article discusses number theoretic multiplicative functions.

## Examples

Examples of multiplicative functions include many functions of importance in number theory, such as:

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*n*,*k*): the greatest common divisor of*n*and*k*, where*k*is a fixed integer. *d*(*n*): the number of positive divisors of*n*,**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma}**(*n*): the sum of all the positive divisors of*n*,**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma}**_{k}(*n*): the divisor function, which is the sum of the*k*-th powers of all the positive divisors of*n*(where*k*may be any complex number). In special cases we have**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma}**_{0}(*n*) =*d*(*n*) and_{1}(*n*) =*n*),

- 1(
*n*): the constant function, defined by 1(*n*) = 1 (completely multiplicative) - Id(
*n*): identity function, defined by Id(*n*) =*n*(completely multiplicative) - Id
_{k}(*n*): the power functions, defined by Id_{k}(*n*) =*n*^{k}for any natural (or even complex) number*k*(completely multiplicative). As special cases we have- Id
_{0}(*n*) = 1(*n*) and - Id
_{1}(*n*) = Id(*n*),

- Id
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon}**(*n*): the function defined by**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon}**(*n*) = 1 if*n*= 1 and = 0 if*n*> 1, sometimes called*multiplication unit for Dirichlet convolution*or simply the*unit function*; sometimes written as*u*(*n*), not to be confused with**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu}**(*n*) (completely multiplicative).- (
*n*/*p*), the Legendre symbol, where*p*is a fixed prime number (completely multiplicative). **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda}**(*n*): the Liouville function, related to the number of prime factors dividing*n*(completely multiplicative).**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma}**(*n*), defined by**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma}**(*n*)=(-1)^{Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} (n)}, where the additive function**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega}**(*n*) is the number of distinct primes dividing*n*.- All Dirichlet characters are completely multiplicative functions.

An example of a non-multiplicative function is the arithmetic function *r*_{2}(*n*) - the number of representations of *n* as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is allowed. For example:

- 1 = 1
^{2}+ 0^{2}= (-1)^{2}+ 0^{2}= 0^{2}+ 1^{2}= 0^{2}+ (-1)^{2}

and therefore *r*_{2}(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, *r*_{2}(*n*)/4 is multiplicative.

In the On-Line Encyclopedia of Integer Sequences, sequences of values of a multiplicative function have the keyword "mult."

See arithmetic function for some other examples of non-multiplicative functions.

## Properties

A multiplicative function is completely determined by its values at the powers of 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}) ...

This property of multiplicative functions significantly reduces the need for computation, as in the following examples for *n* = 144 = 2^{4} · 3^{2}:

- d(144) =
_{0}(144) =_{0}(2^{4})_{0}(3^{2}) = (1^{0}+ 2^{0}+ 4^{0}+ 8^{0}+ 16^{0})(1^{0}+ 3^{0}+ 9^{0}) = 5 · 3 = 15, _{1}(144) =_{1}(2^{4})_{1}(3^{2}) = (1^{1}+ 2^{1}+ 4^{1}+ 8^{1}+ 16^{1})(1^{1}+ 3^{1}+ 9^{1}) = 31 · 13 = 403,^{*}(144) =^{*}(2^{4})^{*}(3^{2}) = (1^{1}+ 16^{1})(1^{1}+ 9^{1}) = 17 · 10 = 170.

Similarly, we have:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi}**(144)=^{4})^{2}) = 8 · 6 = 48

In general, if *f*(*n*) is a multiplicative function and *a*, *b* are any two positive integers, then

Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.

## Convolution

If *f* and *g* are two multiplicative functions, one defines a new multiplicative function *f* * *g*, the *Dirichlet convolution* of *f* and *g*, by

- (
*f***g*)(*n*) = ∑_{d |n}*f*(*d*)*g*(*n*/*d*)

where the sum extends over all positive divisors *d* of *n*.
With this operation, the set of all multiplicative functions turns into an abelian group; the identity element is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon}**
.

Relations among the multiplicative functions discussed above include:

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- Id
_{k}=_{k}*

The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.

## See also

## References

- Tom M.Apostol,
*Introduction to Analytic Number Theory*, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9*(See Chapter 2.)*

de:Multiplikativität es:Función multiplicativa fr:Fonction multiplicative it:Funzione moltiplicativa ko:곱셈적 함수 sv:Multiplikativ funktion zh:積性函數