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

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