We denote the Euler's totient function as $\varphi(n)$ for integers $n\geq 1$, that is a multiplicative function that counts the number of integers $1\leq k\leq n$ up to the given integer n that satisfy $\gcd(k,n)=1$.
I'm curious about if one can to find fixed integers $a\geq 1$ and $b\geq 1$ such that we can prove or refute that the series $$\sum_{n=1}^\infty\frac{1}{n^2+a\varphi(n)+b}$$ is a rational number.
Question. Choose some fixed integers $A\geq 1$ and $B\geq 1$. Can you prove that your series $$\sum_{n=1}^\infty\frac{1}{n^2+A\varphi(n)+B}\tag{1}$$ is a rational number? Or well can (you refute it) you prove that $(1)$ is irrational? Many thanks.
Because I think that this question is very difficult only is required what work can be done about it, and after that there are some answers I should accept one.