Timeline for Behavior of $m(x)\sqrt{x}$ where $m(x)=\sum_{n\leq x}\frac{\mu(n)}{n}$

10 events

when toggle format	what		by	license	comment
Jul 30 at 15:33	comment	added	Babar		@GH from MO Thanks, I expressed myself poorly. I meant to say "the best known result" instead of "tight".
Jul 30 at 15:29	comment	added	GH from MO		@Babar The inequalities by Kotnik & te Riele are not tight. It is expected that $\limsup M(x)x^{-1/2}=\infty$ and $\liminf M(x)x^{1/2}=-\infty$. By following my answer you should be able to find a numeric positive lower bound for $\limsup m(x)x^{1/2}$ and a numeric negative upper bound for $\liminf m(x)x^{1/2}$.
Jul 30 at 15:28	vote	accept	Babar
Jul 30 at 15:25	comment	added	Babar		For the Mertens function $M(x)$, Kotnik & te Riele established: $\limsup M(x)x^{-1/2} > 1.218$, $\liminf M(x)x^{-1/2} < -1.229$. However, analogous tight bounds for $m(x)x^{1/2}$ are unknown to me.
Jul 30 at 15:24	history	edited	GH from MO		edited tags
Jul 30 at 14:56	answer	added	GH from MO		timeline score: 5
Jul 30 at 14:48	comment	added	David E Speyer		See mathoverflow.net/questions/153695 for upper bounds on $\|m(x)\|$, but that doesn't address lower bounds. (I mistakenly thought this was a duplicate of that one, but it isn't.)
Jul 30 at 14:47	history	reopened	David E Speyer nt.number-theory
Jul 30 at 14:46	history	closed	David E Speyer nt.number-theory		Duplicate of Reference and best bounds of $\sum_{n\leq x}\frac{\mu(n)}{n}$
Jul 30 at 14:25	history	asked	Babar	CC BY-SA 4.0