In this post we denote the Gregory coefficients, or reciprocal logarithmic numbers, this Wikipedia Gregory coefficients as $G_k$, for integers $k\geq 1$. I would like to know if it is possible to get upper bounds for next two sequences.
Question. We denote the greatest common factor of two positive integers $a\geq 1$ and $b\geq 1$ as $(a,b)$. Provide the calculations (or hints for one of these similar problems) to get $$\left|\sum_{\substack{1\leq d\leq x\\(d,r)=1}}\frac{G_d}{d^{1+\varepsilon}}\right|\leq \text{upper bound}=\text{upper bound}(\varepsilon,x,r)\tag{1}$$ or for the sequence $$\left|\sum_{\substack{1\leq d\leq x\\(d,r)=1}}\frac{G_d}{d^{1+\varepsilon}}\log\frac{x}{d}\right|\leq \text{upper bound}=\text{upper bound}(\varepsilon,x,r)\tag{2}$$ for reals $\varepsilon>0$ and $x\geq 1$, and integers $r\geq 1$. Many thanks.
Here thus $\text{upper bound}(\varepsilon,x,r)$ denotes a suitable function of $\epsilon$, $x$ and $r$, and I'm asking it (if possible/feasible) in the spirit of Theorem 1.1 and Theorem 1.2 from [1].
As was said if the calculations are similar, provide the reasonings for one of these problems and if you want hints for the other. Any case, I'm asking about what work can be done for some of these problems.
References:
[1] Olivier Ramaré, Some elementary explicit bounds for two mollifications of the Moebius function, Funct. Approx. Comment. Math. Volume 49, Number 2 (2013), pp 229-240.
