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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

  1. has this analogy been exploited already?
  2. are these functions used already?
  3. could one show the orthogonality $$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\overline{\lambda_i(n)}\lambda_j(n)\overset{?}=\delta_{ij}$$
  4. 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.

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  • $\begingroup$ It could be interesting to study the object $ \tilde{\lambda}_{p} : =(\lambda_{p},\lambda_{p^2},\cdots,\lambda_{p^k},\cdots) $ with maybe some extra structure still to be defined. $\endgroup$ – Sylvain JULIEN Dec 26 '17 at 20:24
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    $\begingroup$ The Selberg–Delange method addresses the functions $z^{\omega(n)}$ and $z^{\Omega(n)}$ (indeed, in some sense those are the most natural functions for it to study): one can get an asymptotic formula for their summatory function, which in particular establishes the orthogonality property you suggest. See Tenenbaum's book or Montgomery/Vaughan's book. $\endgroup$ – Greg Martin Dec 26 '17 at 21:58
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    $\begingroup$ What does it mean to speak of an irreducible representation of a function? (I assume also that you mean to speak of the multiplicative 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
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    $\begingroup$ I am afraid that if someone asks "Is it not correct that a multiplicative map between groups is necessarily an isomorphism?" then perhaps they should be asking questions on math.stackexchange.com rather than this site. Moreover, it makes no sense to talk of a representation of $\Omega_n$ in this context. $\endgroup$ – Yemon Choi Dec 27 '17 at 5:35
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    $\begingroup$ I'm voting to close this question as off-topic because it is a cross-post of a question on Math.SE. $\endgroup$ – Stefan Kohl Dec 27 '17 at 11:07

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