The infinite sums involving mobius function and a multiplicative function has got quite interest in past. In particular, sums of the form $$\sum_{d=1}^{\infty}\frac{\mu(d)}{f(d)}$$ for mobius function $\mu$ and multiplicative function $f$ have been investigated for various $f.$ I am interested in knowing about any arguments that could prove/disprove the non-negativity of the following sum $$\sum_{d=1}^{\infty}\frac{\mu(d)}{\mathrm{lcm}(d,\varphi(d))}$$ where $\varphi$ is the euler totient function. The function $f(d)=\mathrm{lcm}(d,\varphi(d))$ is not multiplicative and hence any standard techniques of treating multiplicative $f$ won't work here.

I would like to remark that the sum is absolutely convergent. To see this, one can consider the Lucas sequence $u_n=2^n-1$ and let $\mathrm{ord}_n(2)$ denote the multiplicative order of $2$ modulo $n.$ It is well known that $\mathrm{ord}_n(2)\mid \varphi(n).$ This gives that $$\mathrm{lcm}(n,\varphi(n))\ge \mathrm{lcm}(n,\mathrm{ord}_n(2)).$$ Thus, we have that $$\sum_{d=1}^{\infty}\frac{1}{\mathrm{lcm}(d,\varphi(d))}\le \sum_{d=1}^{\infty}\frac{1}{\mathrm{lcm}(d,\mathrm{ord}_d(2))}$$ and the convergence of right sum follows by proposition 1.4 in this published paper.

Thanks in advance for any help.