Universal sign changes for differences of distribution counts

Prove that, for every pair of distinct admissible vectors \(a\) and \(a'\), the sign of \(|A_a(x)|-|A_{a'}(x)|\) changes infinitely many times as \(x\to\infty\), whenever the leading oscillatory coefficients in the corresponding Dirichlet-series expansion do not cancel.

Background

The proof of Theorem 1 expresses the counting functions through Dirichlet series indexed by roots of unity. For differences ∣Aa(x)∣−∣Aa′(x)∣|A_a(x)|-|A_{a'}(x)|, the dominant common term cancels, leaving lower-order terms whose exponents may be complex.

The paper observes that a nonzero leading coefficient associated with such a complex exponent would produce oscillation in the asymptotic expression and hence infinitely many sign changes. It then conjectures that this nonvanishing behavior holds for every distinct pair, but does not prove it.

References

If given specific a, a', we may directly calculate C, and C\neq 0 would imply the sign of |A_a(x)|-|A_{a'}(x)| changes infinitely many times as x\rightarrow \infty. We conjecture that this is true for any distinct a, a'.

— Distribution of squarefree integers with double congruence conditions  (2609.16716 - Wuji, 15 Sep 2026) in Remark following the proof of Theorem 1, Section 2, and reiterated in Section 6

We conjecture that the sign of |A_a(x)|-|A_a'(x)| changes infinitely many times for all distinct a, a'.

— Distribution of squarefree integers with double congruence conditions  (2609.16716 - Wuji, 15 Sep 2026) in Section 6.2, subsection “\(l_i\geq 3\) for Some \(i\)”