It is known that for each finite set of primes $p$ we have: $\log(p)$ are linear independent over the rational numbers.

We have $\log(ab) = \log(a)+\log(b)$ and $\log(n) = \sum_{p \mid n}v_p(n) \log(p)$.

We define $\psi(n) := \sum_{p \mid n}v_p(n) e_p$ where $e_p$ is the $p$-th standard basis vector of the Hilbert space of sequences.

For $q = a/b \in \mathbb{Q}_{>0}$ we define:

$$\psi(a/b) := \psi(a/{\gcd(a,b)})-\psi(b/{\gcd(a,b)}).$$

We have for $n \in \mathbb{N}$:

$$|\psi(n)|^2 = \sum_{p\mid n} v_p(n)^2 \text{ is a natural number.}$$

Furthermore the function

$$K(a,b) = \left \langle \psi(a),\psi(b)\right \rangle = \sum_{p\mid\gcd(a,b)} v_p(a)v_p(b)$$

is a positive definite kernel on the natural numbers.

Let us define the $\infty$-dimensional lattice $\Gamma$ as:

$\Gamma:= {\psi(q)| q \in \mathbb{Q}_{>0}, |\psi(q)|^2 \equiv 0 \mod(2)}$$

and let us also define:

$$\eta(n) := (-1)^{|\psi(n)|^2}= \prod_{p\mid n}(-1)^{v_p(n)^2}.$$

For $\psi(a),\psi(b) \in \Gamma$ we have:

$$|\psi(ab)|^2 = |\psi(a)+\psi(b)|^2 = |\psi(a)|^2+|\psi(b)|^2+2K(a,b) \equiv 0+0+0 \equiv 0 \mod(2).$$

Therefore $\psi(a)+\psi(b) \in \Gamma$ and $\psi(1) \in \Gamma$.

The lattice shares some properties with the Leech-lattice and the E8 lattice:

It is unimodular in the sense that: For a finite set of primes, the gram matrix $(K(p,q))_{p,q \text{ is prime}}$ has $\det=1$.
It is even: $\forall x \in \Gamma: |x|^2 \equiv 0 \mod(2)$.
For all nonzero vectors in the lattice the squared norm is at least $2$.

My questions are these:

Probably an easy question: $$\eta(mn) = \eta(m) \eta(n) \forall m,n\in \mathbb{N}?$$
Does it satisfy a Riemann-Hypothesis type of random variable equality: $$\forall \epsilon > 0 : \lim_{N\rightarrow \infty} \frac{1}{N^{1/2+\epsilon}}\sum_{n=1}^N \eta(n) = 0?$$
Is there a relationship to the Riemann Hypothesis and this lattice?

And here is some picture concerning the 2. question:

Edit: Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function.

Then because $v_p(n)^2\equiv v_p(n) \mod(2)$, we have indeed:

$$\eta(n)= \lambda(n).$$

This answers the first question since it is known that $\lambda$ is multiplicative.

This also answers the second and third question.

Modified question: Is this lattice known in literature?

Edit: I rewrote some of this for interested readers in this topic here.

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