Editing Openai/68ec50da-cf00-8005-b5f6-b683506e5853 (section)

==== What I changed (and verified) ====
# Complement tail corrected In all bounds for the “complement” block I now use Θ29(2) := 1−∏p≡1 (4)p≥29(1−2p2),\Theta_{29}^{(2)}\ :=\ 1-\prod_{\substack{p\equiv1\ (4)\\ p\ge 29}} \Bigl(1-\frac{2}{p^2}\Bigr),Θ29(2) := 1−p≡1 (4)p≥29∏(1−p22), (two solutions to x2≡−1(modp2)x^2\equiv-1\pmod{p^2}x2≡−1(modp2)).
# Square‑free lemma proof fixed The lemma’s proof now splits primes into three ranges exactly as you proposed: - p>2Np>\sqrt{2N}p>2N: at most one bbb per prime ⇒o(N)\Rightarrow o(N)⇒o(N). - log⁡log⁡N<p≤2N\log\log N<p\le\sqrt{2N}loglogN<p≤2N: ≪N/p2\ll N/p^2≪N/p2 per prime, sum ≪N/log⁡log⁡N\ll N/\log\log N≪N/loglogN. - p≤log⁡log⁡Np\le\log\log Np≤loglogN: genuine CRT/inclusion–exclusion in the progression; yields the Euler product with any forbidden primes (e.g. remove 333 if 3∣a3\mid a3∣a, remove 555 for cross pairs with A7A_7A7 or A18A_{18}A18).
# Compilation fixes - Removed a footnote that had ellipses and could confuse parsers. - Removed non‑LaTeX markup. - Kept notation minimal; all macros are defined in the preamble.