Show that these ratios have numerators $\frac{1}{2} \vartheta(n) (-\exp (\Lambda (n)))$ and denominators $\exp (\Lambda (n))$ for $x=1$,

Question

Let the Dirichlet inverse of the Euler totient function be:

$$\vartheta(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$

and compute the sum:

$$q(x,n)=\sum _{h=0}^{\infty } \left(\sum _{k=1}^n x^{h n+k} \vartheta(\gcd (n,k))\right)$$

This is a sequence of ratios of polynomials:

$$q(x,1,2,3,...)=\frac{x}{1-x},\frac{x}{x+1},\frac{2 x^2+x}{x^2+x+1},\frac{x}{x+1},\frac{4 x^4+3 x^3+2 x^2+x}{x^4+x^3+x^2+x+1},\frac{x-2 x^2}{x^2-x+1},\frac{6 x^6+5 x^5+4 x^4+3 x^3+2 x^2+x}{x^6+x^5+x^4+x^3+x^2+x+1},\frac{x}{x+1},\frac{2 x^2+x}{x^2+x+1},\frac{-4 x^4+3 x^3-2 x^2+x}{x^4-x^3+x^2-x+1},...$$

Separate numerators and denominators, and set $x=1$:

This gives values for numerators of $q(1,n)$:

{1, 1, 3, 1, 10, -1, 21, 1, 3, -2, 55, -1, 78, -3, -4, 1, 136, -1, 171, \
-2, -6, -5, 253, -1, 10, -6, 3, -3, 406, 4,...}

and values for denominators of $q(1,n)$:

{0, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, \
23, 1, 5, 1, 3, 1, 29, 1,...}

https://oeis.org/A014963

Question 1:

Show that numerators are of the form:

$$\frac{1}{2} \vartheta(n) (-\exp (\Lambda (n)))$$

and denominators of the form:

$$\exp (\Lambda (n))$$

Where $\Lambda(n)$ is the von Mangoldt function.

Question 2:

It would also be interesting to know what:

$$\sum _{k=1}^{\infty } y^k q(x,n)$$

converges to.

"Mathematica start:"
Clear[s, x]
a[n_] := Total[Divisors[n]*MoebiusMu[Divisors[n]]]
Monitor[q = 
   Table[Sum[
     Sum[a[GCD[n, k]]*x^(n*h + k), {k, 1, n}], {h, 0, Infinity}], {n, 
     1, 30}];, n]
numerator = Numerator[q]
denominator = Denominator[q]
x = 1;
numerator
denominator
%%/%
"Mathematica end:"

Answer 1

Question 1: $$q(x,n)=-\sum_{d|n} \mu (d)*d*\frac{x^d} {(x^d-1)}=-\frac{d}{dx}\left(\sum_{d|n} \mu (d)*\log(x^d-1) \right)$$ For $n>1$ $$q(x,n)=-\frac{d}{dx}\left(\sum_{d|n} \mu (d)*\log\left(\sum_{l=0}^{d-1}x^l \right) \right)=-\frac{1}{x}\sum_{d|n} \mu (d)*\frac{\sum_{l=1}^{d-1}l*x^l}{\sum_{l=0}^{d-1}x^l}$$ and for $x\rightarrow1$ $$q(1,n)=-\sum_{d|n} \mu (d)*\frac{d-1}{2}=-\frac{\vartheta(n)}{2}$$ So, for any function $f(n)\neq0$ $$q(1,n)=-\frac{\frac{\vartheta(n)}{2}*f(n)}{f(n)}$$ Question 2: For $\vert y\vert<1$ $$\sum_{k=1}^{\infty}y^k*q(x,n)=q(x,n)*\frac{y}{1-y}$$