Asymptotics for a special sum with a divisor function Is there an asymptotic for: $$\sum_{n\leq x\atop n \not\equiv 0(mod\ 2)}d(n)\ \mathrm{\mathbf{and}}\ \sum_{n\leq x\atop n \not\equiv 0(mod\ 2,3)}d(n)$$
I need this for this https://mathoverflow.net/questions/286923/on-a-weak-version-of-the-grimms-conjecture question
Also, if there are any links, I would be only happy.
 A: For the first sum, you are counting pairs $x,y \in \mathbb{N}$ such that $x$ and $y$ are both odd, and $xy \leq X$. You can write this as a sum
$$\displaystyle \sum_{\substack{n \leq X \\ n \text{ odd}}} \sum_{\substack{m \leq X/n \\ m \text{ odd}}} 1.$$
The number of odd integers between 1 and a positive real number $Y$ is $\frac{Y}{2} + O(1)$. Therefore, the inner sum is equal to $\frac{X}{2n} + O(1)$. Now summing $\sum_{\substack{n \leq X \\ n \text{ odd}}}\left( \frac{X}{2n} + O(1)\right)$ gives the asymptotic 
$$\displaystyle \sum_{\substack{n \leq X \\ n \text{ odd}}} \sum_{\substack{m \leq X/n \\ m \text{ odd}}} 1 \sim \frac{X \log X}{4}.$$
For the second case, we shall consider first the sum $\displaystyle \sum_{\substack{n \leq X \\ n \equiv 1 \pmod{6}}} d(n).$ Expanding, this can be written as
$$\displaystyle \sum_{\substack{n \leq X \\ n \equiv 1 \pmod{6}}} \sum_{\substack{m \leq X/n \\ m \equiv 1 \pmod{6}}} 1 + \sum_{\substack{n \leq X \\ n \equiv 5 \pmod{6}}} \sum_{\substack{m \leq X/n \\ m \equiv 5 \pmod{6}}} 1.$$
The evaluation of each sum is similar. In each case, the inner sum evaluates to $\frac{X}{6n} + O(1)$, and then the outer sum evaluates to $\sim \frac{X \log X}{36}$. Thus 
$$\displaystyle \sum_{\substack{n \leq X \\ n \equiv 1 \pmod{6}}} d(n) \sim \frac{X \log X}{18}.$$
Similarly,
$$\displaystyle \sum_{\substack{n \leq X \\ n \equiv 5 \pmod{6}}} d(n) \sim \frac{X \log X}{18}.$$
Therefore the sum you ask for is asymptotic to $(X \log X)/9$. 
