Math 2.0

Number theory Definition

Multiplicative function

Builds on Arithmetic functionCoprime integers

Definition

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

Remarks

A multiplicative function is determined by its values at prime powers: if n=p1e1⋯pkekn = p_1^{e_1} \cdots p_k^{e_k}, then f(n)=f(p1e1)⋯f(pkek)f(n) = f(p_1^{e_1}) \cdots f(p_k^{e_k}). Examples are Euler's totient function, the divisor function and the Möbius function. A function with f(mn)=f(m)f(n)f(mn) = f(m) f(n) for all mm and nn is completely multiplicative.

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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…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

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5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Multiplicative_function, lead section
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Tag0AP80TT660
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Kinddefinition
WritingW1 stated
MSC11A25
DefinesMultiplicative function
UsesArithmetic function, Coprime integers

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