This is a duplicate of a question I have asked at here at math stack exchange, but I thought it could be also here of interest.

When looking at the [Liouville function] (https://en.wikipedia.org/wiki/Liouville_function), defined as $$ \lambda(n) = (-1)^{\Omega_n},$$ where $\Omega_n$ is the total count of prime factors of $n$ (including multiplicity), it occurred to me, that this in a sense parallels an irreducible representation for the multiplicative semigroup of integers since the maps $n\mapsto \Omega_n$ and $n\mapsto\lambda(n)$ are multiplicative, thus homomorphisms. So one could generalise the Liouville function to $$ \lambda_m(n) = (e^{i\frac{2\pi}{m}})^{\Omega_n \mathrm{mod}\,m} ,$$ which recovers for $m=2$ the normal Liouville function $$ \lambda_2(n) = (e^{i\frac{2\pi}{2}})^{\Omega_n \mathrm{mod}\,2} = (-1)^{\Omega_{n} {\mathrm{mod}}\,2} = \lambda(n).$$ In this way one would get other such functions "mimicking" irreducible representations.

The questions are

- has this analogy been exploited already?
- are these functions used already?
- could one show the orthogonality $$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\overline{\lambda_i(n)}\lambda_j(n)\overset{?}=\delta_{ij}$$
- what can be said about the asymptotics of the series $$\sum_{i=1}^{N}\overline{\lambda_i(n)}\lambda_i(m)$$ which would correspond to the "column orthogonality" in the character table, and be the order of the centraliser of the (semi?)group for conjugated elements $n,m$ and be $0$ otherwise?

For example for $\lambda_1(n):=1=\lambda_j(n)\;\forall n$ (the "totally symmetric representation") and $\lambda_i(n)=\lambda(n)$, that reduces to $$\lim_{N\rightarrow\infty}\frac{L_N}{N},$$ with the Liouville sum function $L_N$. The statement that this limit $=0$ is apparently equivalent to the prime number theorem (while $\lim_{N\rightarrow\infty}\frac{L_N}{N^{\frac{1}{2}+\epsilon}}=0$ for any $\epsilon>0$ is RH). I do not have any experiences with groups of infinte order, but the question if $\lim_{N\rightarrow\infty}\frac{\sum_{n=1}^{N}\lambda_i(n)}{\sqrt{N}}$ or similar converges, seems closely connected with the question if $\lambda_i$ can be normalised and hence if they are "proper" irreducible representations distributed evenly around the unit circle.

semigroup of integers. Ah, one more: I should also say that $n \mapsto \Omega_n$ is certainly multiplicative, but not, obviously, an isomorphism.) $\endgroup$ – LSpice Dec 26 '17 at 23:42