Let $M(x)$ be the Mertens function. Let us define the function $n(j)=2+\sum_{i=1}^{j}|sgn(M(i))|$ over natural numbers $j$ and the function $s_j(x)=-\sum_{i=1}^{\lfloor\frac{x}{j+1}\rfloor}sgn\left(M\lfloor\frac{x}{i}\rfloor\right)\cdot i$
Why the following inequality holds for natural numbers $x>1$?
$\frac{1}{2}\cdot\lfloor\frac{x}{n(j)}\rfloor\lceil\frac{x}{n(j)}\rceil\ge s_j(x)$
Empirically, when choosing for example $j=100$ and we let $x$ running from $2$ up to $1000$ ($x=2,3,4,\ldots,1000$), this inequality seems to be true:
