发表机构
Pontificia Universidad Javeriana Cali（波哥大耶稣会天主教大学卡利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对模6P的偶数受限互素表示，利用容斥原理和中国剩余定理给出至多3^r项的精确公式，证明仿射恒等式及有限密度分解，无需概率独立性假设。
AI 中文摘要
设 $p_1,\dots,p_r\geq 5$ 为互不相同的素数，令 $P=p_1\cdots p_r$，并置 $M=6P$。对于正整数 $n$，记 $g_P(2n)$ 为无序表示 $2n=h+k$ 的个数，其中 $1\leq h\leq k$，且满足 $\gcd(h,M)=\gcd(k,M)=1$。利用典范余数算子 $\delta_q(x)=x-q\lfloor x/q\rfloor$，我们识别出每个素数 $p_i$ 所排除的两个局部剩余类。它们的可能碰撞由 $\delta_{p_i}(n)=0$ 精确刻画。结合容斥原理与中国剩余定理，得到一个至多含 $3^r$ 项的精确公式。我们证明相关的有限关联的基数为 $\kappa_P(n)=(1+\mathbf{1}_{\{\delta_3(n)=0\}})\prod_{i=1}^{r}(p_i-2+\mathbf{1}_{\{\delta_{p_i}(n)=0\}})$，并获得仿射恒等式 $g_P(2(n+6P))-g_P(2n)=\kappa_P(n)$。我们还建立了精确的有限密度分解，其一致误差小于 $3^r-1$，一个半周期正性定理，以及一个乘积截断准则。最后，我们阐述了该问题的覆盖与配对间隙解释，并精确识别出当所有不超过 $\sqrt{2n}$ 的素数被纳入时出现的确定性边界。未使用任何概率独立性假设。
英文摘要
Let $5\leq p_1<\cdots<p_r$ be primes, put $P=p_1\cdots p_r$, and let $g_P(2n)$ count the unordered decompositions $2n=h+k$ for which both summands are coprime to $6P$. We derive an explicit finite inclusion--exclusion formula for $g_P(2n)$ by parametrizing the admissible residue classes and combining the local restrictions through the Chinese remainder theorem. The associated complete-modulus correlation is a specialization of Nagell's totient function and determines both the finite local density and the affine increment $g_P(2(n+6P))-g_P(2n)=κ_P(n)$. A parameter-density decomposition gives a residue-sensitive positivity criterion and the uniform sufficient condition $n\geq 3(3^r-1)\prod_{i=1}^{r}\frac{p_i}{p_i-2}+2\Longrightarrow g_P(2n)>0$. Maximizing this bound over squarefree moduli whose prime factors do not exceed a prime $p$ gives an explicit threshold $\mathcal N(p)$ satisfying $\log\mathcal N(p)\sim(\log3)p/\log p$. We also describe the complete prime cutoff at $\sqrt{2n}$ and its exact additive-prime interpretation. We also evaluate $g_P(2tP)$ explicitly using the real character of period $3$. The positivity thresholds are elementary and fully explicit; lower-bound sieves provide smaller asymptotic scales, and the present thresholds do not establish positivity at the complete cutoff for all $n\geq13$.
Comments25 pages, no figures. Revised title and reorganized exposition. Added refined discrepancy bounds, a half-period identity, explicit positivity thresholds, and a comparison with lower-bound sieves. Strengthened the complete-cutoff characterization to an exact counting identity. Updated references
