non-multiplicative function


Examples

Some examples of a non-multiplicative functions are the arithmetic functions:

  • r2(n) - the number of unordered representations of n as a sum of squares of two integers, positive, negative or zero,

  • c4(n) - the number of ways that n can be expressed as the sum of four squares of nonnegative integers, where we distinguish between different orders of the summands. For example:

    1=12+02+02+02=02+12+02+02+02=02+02+12+02=02+02+02+12,

    hence c4(1)=41.

  • The partition function P(n) - the number of ordered representations of n as a sum of positive integers. For instance:

    P(25)=P(10)=42and
    P(2)P(5)=27=1442.
  • The prime counting function π(n). Here we first have π(1)=01 and then we have as yet for example:

    π(25)=π(10)=4and
    π(2)π(5)=13=34.
  • The Mangoldt functionDlmfMathworldPlanetmath Λ(n). Λ(1)=ln11 and for example:

    Λ(25)=Λ(10)=0and
    Λ(2)Λ(5)=ln2ln50.

    We would think that for some n multiplicativity of Λ(n) would be true as in:

    Λ(26)=Λ(12)=0and
    Λ(2)Λ(6)=ln20=0,

    but we have to write:

    Λ(22)Λ(3)=ln2ln30.
Title non-multiplicative function
Canonical name NonmultiplicativeFunction
Date of creation 2013-03-22 12:47:04
Last modified on 2013-03-22 12:47:04
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 16
Author Mathprof (13753)
Entry type Example
Classification msc 11A25
Related topic PartitionFunction2
Defines partition function