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It is known to be false that $\sum_{m\leq x} \mu(m) \leq \sqrt{x}$ for all $x$ (Mertens' conjecture), and it is generally believed that $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$. From the latter, it would follow that $$\lim \sup_{x\to\infty} \sqrt{x} \left|\sum_{m\leq x} \frac{\mu(m)}{m}\right| = \infty,$$ by partial summation. However, what about a smoothed sum, such as $$\sum_{m\leq x} \frac{\mu(m)}{m} \log \frac{x}{m}?$$ Is it clear that $$\lim \sup_{x\to\infty} \sqrt{x} \left|\sum_{m\leq x} \frac{\mu(m)}{m} \log \frac{x}{m} \right| = \infty?$$ If one can't deduce the truth of this easily from $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$, I'd be interested in what standard random models imply on the matter.

It is known to be false that $\sum_{m\leq x} \mu(m) \leq \sqrt{x}$ for all $x$ (Mertens' conjecture), and it is generally believed that $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$. From the latter, it would follow that $$\lim \sup_{x\to\infty} \sqrt{x} \left|\sum_{m\leq x} \frac{\mu(m)}{m}\right| = \infty,$$ by partial summation. However, what about a smoothed sum, such as $$\sum_{m\leq x} \frac{\mu(m)}{m} \log \frac{x}{m}?$$ Is it clear that $$\lim \sup_{x\to\infty} \sqrt{x} \left|\sum_{m\leq x} \frac{\mu(m)}{m} \log \frac{x}{m} - 1\right| = \infty?$$ If one can't deduce the truth of this easily from $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$, I'd be interested in what standard random models imply on the matter.

It is known to be false that $\sum_{m\leq x} \mu(m) \leq \sqrt{x}$ for all $x$ (Mertens' conjecture), and it is generally believed that $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$. From the latter, it would follow that $$\lim \sup_{x\to\infty} \sqrt{x} \left|\sum_{m\leq x} \frac{\mu(m)}{m}\right| = \infty,$$ by partial summation. However, what about a smoothed sum, such as $$\sum_{d\leq x} \frac{\mu(m)}{m} \log \frac{x}{m}?$$ Is it clear that $$\lim \sup_{x\to\infty} \sqrt{x} \left|\sum_{m\leq x} \frac{\mu(m)}{m} \log \frac{x}{m} \right| = \infty?$$ If one can't deduce the truth of this easily from $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$, I'd be interested in what standard random models imply on the matter.

It is known to be false that $\sum_{m\leq x} \mu(m) \leq \sqrt{x}$ for all $x$ (Mertens' conjecture), and it is generally believed that $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$. From the latter, it would follow that $$\lim \sup_{x\to\infty} \sqrt{x} \left|\sum_{m\leq x} \frac{\mu(m)}{m}\right| = \infty,$$ by partial summation. However, what about a smoothed sum, such as $$\sum_{m\leq x} \frac{\mu(m)}{m} \log \frac{x}{m}?$$ Is it clear that $$\lim \sup_{x\to\infty} \sqrt{x} \left|\sum_{m\leq x} \frac{\mu(m)}{m} \log \frac{x}{m} \right| = \infty?$$ If one can't deduce the truth of this easily from $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$, I'd be interested in what standard random models imply on the matter.