# non-multiplicative function

## 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 $c_{4}(1)=4\neq 1\;.$

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

 $P(2\cdot 5)=P(10)=42\quad\hbox{and}$
 $P(2)P(5)=2\cdot 7=14\neq 42\;.$
• The prime counting function $\pi(n)$. Here we first have $\pi(1)=0\neq 1$ and then we have as yet for example:

 $\pi(2\cdot 5)=\pi(10)=4\quad\hbox{and}$
 $\pi(2)\pi(5)=1\cdot 3=3\neq 4\;.$
• The Mangoldt function $\Lambda(n)$. $\Lambda(1)=\ln 1\neq 1$ and for example:

 $\Lambda(2\cdot 5)=\Lambda(10)=0\quad\hbox{and}$
 $\Lambda(2)\Lambda(5)=\ln 2\cdot\ln 5\neq 0\;.$

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:

 $\Lambda(2^{2})\Lambda(3)=\ln 2\cdot\ln 3\neq 0\;.$
Title non-multiplicative function NonmultiplicativeFunction 2013-03-22 12:47:04 2013-03-22 12:47:04 Mathprof (13753) Mathprof (13753) 16 Mathprof (13753) Example msc 11A25 PartitionFunction2 partition function