Exact Formulas for Restricted Coprime Representations of Even Integers with Squarefree Modulus 6P

Abstract

Let $p_1,\dots,p_r\geq 5$ be distinct primes, let $P=p_1\cdots p_r$, and put $M=6P$. For a positive integer $n$, let $g_P(2n)$ denote the number of unordered representations $2n=h+k$, with $1\leq h\leq k$, such that $\gcd(h,M)=\gcd(k,M)=1$. Using the canonical remainder operator $δ_q(x)=x-q\lfloor x/q\rfloor$, we identify the two local residue classes excluded by each prime $p_i$. Their possible collision is characterized exactly by $δ_{p_i}(n)=0$. An inclusion--exclusion argument combined with the Chinese remainder theorem gives an exact formula containing at most $3^r$ terms. We prove that the associated finite correlation has cardinality $κ_P(n)=(1+\mathbf{1}_{\{δ_3(n)=0\}})\prod_{i=1}^{r}(p_i-2+\mathbf{1}_{\{δ_{p_i}(n)=0\}})$ and obtain the affine identity $g_P(2(n+6P))-g_P(2n)=κ_P(n)$. We also establish an exact finite-density decomposition with uniform error smaller than $3^r-1$, a half-period positivity theorem, and a product-cutoff criterion. Finally, we formulate the covering and paired-gap interpretations of the problem and identify precisely the deterministic boundary that appears when all primes not exceeding $\sqrt{2n}$ are included. No probabilistic independence assumption is used.