$\displaystyle \sum_{n \le x} y^{\Omega(n)}=O\left( \frac{x(\log x)^{y-1}}{2-y} \right)$ for $1 \le y<2$

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Type of Math Object:
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Would someone mind proofreading this and see if the range on y can be extended further than $1 \le y < 2$? (The maximum is $0 \le y <2$, which seems to be possible.) I would greatly appreciate it. Thanks.