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: $$ P(2 \cdot 5) = P(10) = 42 \quad \hbox{and} $$ $$ P(2) P(5) = 2 \cdot 7 = 14 \ne 42 \; . $$
• The prime counting function $\pi(n)$ . Here we first have $\pi(1) = 0 \ne 1$ and then we have as yet for example: $$ \pi(2 \cdot 5) = \pi(10) = 4 \quad \hbox{and} $$ $$ \pi(2) \pi(5) = 1 \cdot 3 = 3 \ne 4 \; . $$
• The Mangoldt function $\Lambda(n)$ . $\Lambda(1) = \ln 1 \ne 1$ and for example: $$ \Lambda(2 \cdot 5) = \Lambda(10) = 0 \quad \hbox{and} $$ $$ \Lambda(2) \Lambda(5) = \ln 2 \cdot \ln 5 \ne 0 \; . $$ We would think that for some $n$ multiplicativity of $\Lambda(n)$ would be true as in: $$ \Lambda(2 \cdot 6) = \Lambda(12) = 0 \quad \hbox{and} $$ $$ \Lambda(2) \Lambda(6) = \ln 2 \cdot 0 = 0 \; , $$ but we have to write: $$ \Lambda(2^{2}) \Lambda(3) = \ln 2 \cdot \ln 3 \ne 0 \; . $$