PlanetMath: non-multiplicative function

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non-multiplicative function

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

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

$r_{2}(n)$ - the number of unordered representations of as a sum of squares of two integers, positive, negative or zero,
$c_{4}(n)$ - the number of ways that can be expressed as the sum of four squares of nonnegative integers, where we distinguish between different orders of the summands. For example:
$\displaystyle 1 = 1^{2}+0^{2}+0^{2}+0^{2} = 0^{2}+1^{2}+0^{2}+0^{2}+0^{2} = 0^{2}+0^{2}+1^{2}+0^{2} = 0^{2} + 0^{2} + 0^{2} + 1^{2} \; ,$
hence $c_{4}(1)=4 \ne 1 \; .$
The partition function - the number of ordered representations of as a sum of positive integers. For instance:
$\displaystyle P(2 \cdot 5) = P(10) = 42 \quad \hbox{and}$

$\displaystyle P(2) P(5) = 2 \cdot 7 = 14 \ne 42 \; .$
The prime counting function $\pi(n)$ . Here we first have $\pi(1) = 0 \ne 1$ and then we have as yet for example:
$\displaystyle \pi(2 \cdot 5) = \pi(10) = 4 \quad \hbox{and}$

$\displaystyle \pi(2) \pi(5) = 1 \cdot 3 = 3 \ne 4 \; .$
The Mangoldt function $\Lambda(n)$ . $\Lambda(1) = \ln 1 \ne 1$ and for example:
$\displaystyle \Lambda(2 \cdot 5) = \Lambda(10) = 0 \quad \hbox{and}$

$\displaystyle \Lambda(2) \Lambda(5) = \ln 2 \cdot \ln 5 \ne 0 \; .$
We would think that for some multiplicativity of $\Lambda(n)$ would be true as in:
$\displaystyle \Lambda(2 \cdot 6) = \Lambda(12) = 0 \quad \hbox{and}$

$\displaystyle \Lambda(2) \Lambda(6) = \ln 2 \cdot 0 = 0 \; ,$
but we have to write:
$\displaystyle \Lambda(2^{2}) \Lambda(3) = \ln 2 \cdot \ln 3 \ne 0 \; .$

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