The Fourier transform of the Liouville function?

Question

The Liouville function in number theory is defined as:

$$\lambda(n) := (-1)^{\Omega(n)} \text{ where } \Omega(n) := \sum_{p|n} v_p(n)$$

Taking the discrete time Fourier transform and then taking the inverse, I arrived at the following function for $\lambda$ extended to $\mathbb{R}$ or $\mathbb{C}$:

$$\lambda(x) = \sum_{z \in \mathbb{Z}} \frac{\cos(\pi(\Omega(z)+x-z))\sin(\pi(x-z))}{\pi(x-z)}$$

where if $x$ is an integer $a$ the limit for $x \rightarrow a$ gives $\lambda(a) = (-1)^{\Omega(a)}$. Here is an image where red points correspond to $(n,\lambda(n))$ and blue lines correspond to $\lambda(n)$ whereas the green function corresponds to the extended function $\lambda(x)$:

This can also be written as:

$$\lambda(x) = \frac{\cos(\pi x)\sin(\pi x)}{\pi}\gamma(x)$$

where $\gamma(x)$ is defined as:

$$\gamma(x) = \sum_{z \in \mathbb{Z}} \frac{(-1)^{\Omega(z)}}{x-z}$$ $$=\frac{1}{x}+2x\sum_{n=1}^{\infty} \frac{(-1)^{\Omega(n)}}{x^2-n^2}$$

I tried to search for "extension of Liouville function" or "liouville function" "fourier transform" but could not find anything similar to this representations.

Q1) Is the function $\gamma(x)$ something known or can it be transformed to something more familiar (It looks a little bit like the Weierstraß elliptic function but I am not sure what this means)?

Q2) Is there a differential equation which is satisfied by the $\lambda$ function or the $\gamma$ function?

For Q2), we have the following differential equation:

$$(\frac{\lambda(x)}{\gamma(x)})'' = -4 \pi^2 \frac{\lambda(x)}{\gamma(x)}$$

but this is not really useful, since we have two rather unknown functions popping up in the differential equation. It would be nice, if it is possible to give a differential equation where only terms of one function appear.

As a side note, if it is of interest:

Using the known Dirichlet series of $(-1)^{\Omega(n)}$ one can derive for $0 < |x| < 1$ the following series:

$$\gamma(x) = \frac{1}{x} - 2 \sum_{n=0}^{\infty} \frac{\zeta(4(n+1))}{\zeta(2(n+1))}x^{2n+1}$$

Here is a plot of the gamma function in the complex numbers:

Thanks for your help.

Edit: Here is the Sagemath code to do the example computations which shows that in both cases for $x=0$ we have $\lambda(x)=1$.

Application:

Denote with $\beta(x) = \sum_{n=1}^{\infty} \frac{(-1)^{\Omega(n)}}{x^2-n^2}$, so that $\gamma(x) = 1/x+2x\beta(x)$.

Then we have the following functional equation:

$$\forall a,b \in \mathbb{Z}: \lambda(a+b) = \lambda(a) \lim_{x\rightarrow a}\frac{\gamma(x+b)}{\gamma(x)}$$

From this one deduces:

1)

$$\forall n \in \mathbb{Z}: \lambda(n+1) = \lambda(n) \lim_{x\rightarrow n}\frac{\gamma(x+1)}{\gamma(x)}$$

2)

$$\forall a,b \in \mathbb{Z}: \lambda(ab) = \lim_{x\rightarrow a+b}\frac{\gamma(x)^2}{\gamma(x-a)\gamma(x-b)}$$

3)

Then for a normalized polynomial $P(x) = \prod_{i} (x-\alpha_i)$ with integer roots $\alpha_i$ we get as an application, where $\Lambda = $ denotes the set of roots:

$$\forall x \in \mathbb{Z}, x \notin \Lambda:$$

$$\lambda(P(x)) = \lim_{t \rightarrow x} \frac{\lambda(t)^{\operatorname{deg}(P)}}{(1/t+2t\beta(t))^{\operatorname{deg}(P)}}\prod_{\alpha \in \Lambda} (1/(t-\alpha)+2(t-\alpha)\beta(t-\alpha))$$

Here is an example computation in SageMath.

Answer 1

Your first formula for $\lambda(x)$ is equivalent to

$$\lambda(x)=\underset{N\to \infty}{\text{lim}}\left(\sum\limits_{z=-N}^N \cos(\pi(\Omega(z)+x-z))\, \text{sinc}(\pi(x-z))\right)\tag{1}.$$

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Another way to evaluate $\lambda(x)$ is

$$\lambda(x)=\frac{1}{2} \underset{N\to \infty}{\text{lim}}\left(\sum\limits_{n=1}^N \frac{\lambda(n)\, 2^{\nu(n)}}{n} \left(1-2 n\, \text{sinc}(2 \pi x)+\cos(2 \pi x)+2 \sum\limits_{k=1}^{n-1} \cos\left(\frac{2 \pi k x}{n}\right)\right)\\=\frac{1}{2} \underset{N\to \infty}{\text{lim}}\left(\sum\limits_{n=1}^N \frac{\lambda(n)\, 2^{\nu(n)}}{n} \left(\sin(2 \pi x) \cot\left(\frac{\pi x}{n}\right)-2 n\, \text{sinc}(2 \pi x)\right)\right)\right)\tag{2}$$

where $\nu(n)$ is the number of distinct primes dividing $n$.

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Formula (2) above is based on this answer I posted to a question on an entire function interpolating the Möbius function $\mu(n)$, and likewise I believe formula (2) above is an entire function interpolating the Liouville function $\lambda(n)$.

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Figure (1) below illustrates formula (2) above in orange overlaid on formula (1) in blue where both formulas are evaluated using the upper evaluation limit $N=20$. The red discrete points in Figure (1) below illustrate the evaluation of $\lambda(|x|)$ at integer values of $x$.

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Figure (1): Illustration of formulas (1) and (2) for $\lambda(x)$ in orange and blue respectively

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Figure (2) below illustrates that the formula

$$\lambda(x)=\frac{\cos(\pi x)\, \sin(\pi x)}{\pi}\, \gamma(x)\tag{3}$$

which is the second formula for $\lambda(x)$ in the question above where

$$\gamma(x)=\frac{1}{x}+2 x\, \underset{N\to \infty}{\text{lim}}\left(\sum\limits_{n=1}^N \frac{(-1)^{\Omega(n)}}{x^2-n^2}\right)\tag{4}$$

which is equivalent to the second formula for $\gamma(x)$ in the question above evaluates to one at $x=0$, whereas formula (1) above which is equivalent to the first formula for $\lambda(x)$ in the question above evaluates to zero at $x=0$ as illustrated in Figure (1) above.

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Figure (2): Illustration of Formula (3) where the underlying Formula (4) is evaluated at $N=20$

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