# P11562: Local Prime-Exchange Balancing and Möbius Cancellation

- ID: `P11562`
- Reference: `local-prime-exchange-balancing-mobius-cancellation`
- Page: https://theoremdb.org/statements/P11562
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/local-prime-exchange-balancing-mobius-cancellation?representation=markdown
- Record maturity: Reviewed problem

## The problem

For \(x\ge 1\), let
\[
\mathcal S_x=\{n\le x:n\text{ is squarefree}\}.
\]
For \(n\in\mathcal S_x\), write \(P(n)\) for its set of prime divisors, so
\[
\mu(n)=(-1)^{|P(n)|}.
\]
Let
\[
\mathcal S_x^+=\{n\in\mathcal S_x:\mu(n)=+1\},\qquad
\mathcal S_x^-=\{n\in\mathcal S_x:\mu(n)=-1\}.
\]
Define a bipartite graph \(G_x\) on \(\mathcal S_x\) by joining \(m\) and \(n\) exactly when
\[
|P(m)\triangle P(n)|\in\{1,3\}.
\]
Thus every edge joins opposite Möbius signs. Set \(K_0(m,n)=1\) on edges and \(0\) otherwise. Given a nonnegative symmetric edge kernel \(K_j\), define
\[
r_j(n)=\sum_{m\sim n}K_j(n,m),
\]
and, whenever all relevant row masses are positive, define
\[
K_{j+1}(m,n)=\frac{K_j(m,n)}{\sqrt{r_j(m)r_j(n)}}
\]
on edges.

Prove that for every \(\varepsilon>0\) there exist explicit constants \(C_\varepsilon,A_\varepsilon>0\) and an explicit integer-valued function \(J(x)\) satisfying
\[
J(x)\le C_\varepsilon(\log x)^{A_\varepsilon}
\]
such that, for all sufficiently large \(x\),
\[
\sum_{n\in\mathcal S_x}|r_{J(x)}(n)-1|
\le C_\varepsilon x^{1/2+\varepsilon}.
\]
The proof must derive this bound from the local arithmetic geometry of \(G_x\) and the stated balancing dynamics, without using estimates for the Mertens function \(M(x)\), locations of zeros of \(\zeta\), or a separately optimized global matching.

Since \(K_j\) is symmetric and every edge joins opposite signs,
\[
\sum_{n\in\mathcal S_x^+}r_j(n)=\sum_{n\in\mathcal S_x^-}r_j(n).
\]
Hence
\[
|M(x)|\le \sum_{n\in\mathcal S_x}|1-r_j(n)|.
\]
Therefore the target estimate implies
\[
M(x)=O_\varepsilon(x^{1/2+\varepsilon}),
\]
and hence the Riemann Hypothesis.

As an intermediate structural target, prove or refute that \(G_x\) has a matching saturating the smaller of \(\mathcal S_x^+\) and \(\mathcal S_x^-\) 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.

## Status

**Resolution status:** Open.
**Lean verification:** Not Lean-verified.

A Lean formalization is attached. Signed verification is pending.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `local-prime-exchange-balancing-mobius-cancellation`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.