Residue Sums Modulo Products of Two Distinct Odd Primes

Source abstract

Let $S_m(a)$ denote the sum of the least positive residues in one multiplicative period of $a$ modulo $m$, of length $k$. We study the equality $S_m(a)=mk/2$ for $m=pq$, where $p,q$ are distinct odd primes and $k$ is even. Complementary pairing settles subgroups containing $-1$; the remaining cases depend on their embedding in the two local unit groups. At index four, we exclude mixed cyclic subgroups with biquadratic annihilator and classify every eligible imprimitive quartic kernel using two odd integer moments at its prime conductor. For each fixed conductor, the proportion of admissible second primes giving equality is $1/4$ or $1/2$; full coset balance has proportion $1/4$. For primitive quartic characters, classical triangular character-sum identities yield explicit short-sum criteria, and the relative class-number formula gives an obstruction to a centered coset. At index six, we give structural counts and construct an infinite family modulo $19q$ in which nonzero sextic and quadratic moments cancel. These applications use a standard odd-character expansion, which also recovers the known index-two balance classification and determines the sign of its deviation. A reproducible census for $pq\le10000$ counts $75\,090$ eligible cyclic subgroups and identifies $101$ mixed index-four subgroups whose own sum is centered although some coset sums are not.