non-multiplicative function

In number theory, a non-multiplicative function is an arithmetic function which is not multiplicative.

Examples

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

• $r_2(n)$ - the number of unordered representations of $n$ as a sum of squares of two integers, positive, negative or zero,

• $c_4(n)$ - 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:

  $$1=1^2+0^2+0^2+0^2=0^2+1^2+0^2+0^2+0^2=0^2+0^2+1^2+0^2=0^2+0^2+0^2+1^2,$$

  hence $c_4(1)=4\ne 1.$

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

  $$P(2\cdot 5)=P(10)=42\mathit{\hspace{1em}}\text{and}$$

  $$P(2)P(5)=2\cdot 7=14\ne 42.$$

• The prime counting function $\pi (n)$. Here we first have $\pi (1)=0\ne 1$ and then we have as yet for example:

  $$\pi (2\cdot 5)=\pi (10)=4\mathit{\hspace{1em}}\text{and}$$

  $$\pi (2)\pi (5)=1\cdot 3=3\ne 4.$$

• The Mangoldt function $\mathrm{\Lambda }(n)$. $\mathrm{\Lambda }(1)=\ln 1\ne 1$ and for example:

  $$\mathrm{\Lambda }(2\cdot 5)=\mathrm{\Lambda }(10)=0\mathit{\hspace{1em}}\text{and}$$

  $$\mathrm{\Lambda }(2)\mathrm{\Lambda }(5)=\ln 2\cdot \ln 5\ne 0.$$

  We would think that for some $n$ multiplicativity of $\mathrm{\Lambda }(n)$ would be true as in:

  $$\mathrm{\Lambda }(2\cdot 6)=\mathrm{\Lambda }(12)=0\mathit{\hspace{1em}}\text{and}$$

  $$\mathrm{\Lambda }(2)\mathrm{\Lambda }(6)=\ln 2\cdot 0=0,$$

  but we have to write:

  $$\mathrm{\Lambda }(2^2)\mathrm{\Lambda }(3)=\ln 2\cdot \ln 3\ne 0.$$