In this post we denote the Gregory coefficients, or reciprocal logarithmic numbers, this Wikipedia [*Gregory coefficients*](https://en.wikipedia.org/wiki/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 integer $r\geq 1$. **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. 

References:
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[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.