Is it possible to find an estimate of the summation
$$s(n)=\sum\_{k=1}^n\frac1{\varphi(k\cdot p\_k)}$$
where $\varphi(n)$ is the totient function and $p\_k$ the k-th prime?


The corresponding series seems to converge to the value
$$\lim\_{n\rightarrow\infty}s(n)=1.86491\ldots$$
but I don't see a simple way to prove it.


Many thanks.



Is it possible to find an estimate of $\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$?
$\newcommand\vpi\varphi$[It is known](https://en.wikipedia.org/wiki/Euler%27s_totient_function#Growth_rate) that
$$\vpi(n)\ge\frac n{c\ln\ln(n+10)}=:\psi(n)$$
for some real $c>0$ and all natural $n$. Also, $\psi$ is increasing on the interval $[N,\infty)$ for some natural $N$.
Therefore and because $p\_k\ge k$ for all $k$,
$$\sum\_{k\ge N}\frac1{\vpi(kp\_k)}\le\sum\_{k\ge N}\frac1{\psi(kp\_k)}
\le\sum\_{k\ge N}\frac1{\psi(k^2)}<\infty.$$


