Math 2.0

Number theory Definition

Completely multiplicative function

Builds on Arithmetic functionMultiplicative function

Definition

An arithmetic function ff is completely multiplicative if f(1)=1f(1) = 1 and f(mn)=f(m)f(n)f(mn) = f(m) f(n) for all positive integers mm and nn.

Remarks

Each completely multiplicative function is multiplicative. The converse fails: Euler's totient function has φ(4)=2\varphi(4) = 2 but φ(2)2=1\varphi(2)^2 = 1. A completely multiplicative function is determined by its values at the primes. Examples are n↦nkn \mapsto n^k, the Liouville function and, for a fixed odd prime pp, the Legendre symbol n↦(np)n \mapsto \left(\frac{n}{p}\right).

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Arithmetic function, definition, no evidence levelArithmetic functiondefinitionDivisibility, definition, no evidence levelDivisibilitydefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionCoprime integers, definition, no evidence levelCoprime integersdefinitionMultiplicative function, definition, no evidence levelMultiplicative functi…definitionCompletely multiplicative function, definition, no evidence levelCompletely multiplica…definition
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  1. Arithmetic function definition
  2. Divisibility definition
  3. Greatest common divisor definition
  4. Coprime integers definition
  5. Multiplicative function definition
  6. Completely multiplicative function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Completely_multiplicative_function, Definition
About this page
Tag0GS27TKF1V
URIm2://0GS27TKF1V
Aliasestotally-multiplicative-function
Kinddefinition
WritingW1 stated
MSC11A25
DefinesCompletely multiplicative function
UsesArithmetic function, Multiplicative function

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