${\displaystyle \sum \mathrm{@}\mathrm{@}\mathrm{@}{\mathrm{@}}_{n\le x}{y}^{\omega (n)}}=O_y(x{(logx)}^{y-1})$ for $y\ge 0$

Within this entry, $\omega$ refers to the number of distinct prime factors function, $\lfloor \cdot \rfloor$ refers to the floor function, $\log$ refers to the natural logarithm, $p$ refers to a prime, and $k$ and $n$ refer to positive integers.

Theorem 1.

For $y\mathrm{\ge }\mathrm{0}$, ${\displaystyle \mathrm{\sum }_{n\mathrm{\le }x}}{y}^{\omega \mathit{}\mathrm{(}n\mathrm{)}}\mathrm{=}{O}_y\mathit{}\mathrm{(}x\mathit{}{\mathrm{(}\log \mathit{}x\mathrm{)}}^{y\mathrm{-}\mathrm{1}}\mathrm{)}$.

Proof.

Since $y^{\omega (p^k)}=y$ for all $p$ and $k$, the real-valued nonnegative multiplicative function $y^{\omega (n)}$ the Wirsing condition with $c=y$ and $\lambda =1$. Thus:

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