For x≥1, let
Sx={n≤x:n is squarefree}.
For n∈Sx, write P(n) for its set of prime divisors, so
μ(n)=(−1)∣P(n)∣.
Let
Sx+={n∈Sx:μ(n)=+1},Sx−={n∈Sx:μ(n)=−1}.
Define a bipartite graph Gx on Sx by joining m and n exactly when
∣P(m)△P(n)∣∈{1,3}.
Thus every edge joins opposite Möbius signs. Set K0(m,n)=1 on edges and 0 otherwise. Given a nonnegative symmetric edge kernel Kj, define
rj(n)=m∼n∑Kj(n,m),
and, whenever all relevant row masses are positive, define
Kj+1(m,n)=rj(m)rj(n)Kj(m,n)
on edges.
Prove that for every ε>0 there exist explicit constants Cε,Aε>0 and an explicit integer-valued function J(x) satisfying
J(x)≤Cε(logx)Aε
such that, for all sufficiently large x,
n∈Sx∑∣rJ(x)(n)−1∣≤Cεx1/2+ε.
The proof must derive this bound from the local arithmetic geometry of Gx and the stated balancing dynamics, without using estimates for the Mertens function M(x), locations of zeros of ζ, or a separately optimized global matching.
Since Kj is symmetric and every edge joins opposite signs,
n∈Sx+∑rj(n)=n∈Sx−∑rj(n).
Hence
∣M(x)∣≤n∈Sx∑∣1−rj(n)∣.
Therefore the target estimate implies
M(x)=Oε(x1/2+ε),
and hence the Riemann Hypothesis.
As an intermediate structural target, prove or refute that Gx has a matching saturating the smaller of Sx+ and Sx− for every x, equivalently that its matching defect is exactly ∣M(x)∣. More generally, determine the least fixed odd exchange radius r for which an analogous local balancing mechanism can achieve square-root-scale defect.