QSS: Exact and Accelerated Evaluation of a Sum of Products of Integer Quotients

This paper considers the exact evaluation of the arithmetic sum of products of integer quotients G(N) for a positive integer parameter N. We propose and investigate an implementation called QSS (Quotient-Split Summation), which combines direct evaluation of an initial segment with block grouping of indices by equal values of the first integer quotient. For each block, we prove that the second quotient takes at most two distinct consecutive values, allowing the block contribution to be computed analytically using a small, constant number of arithmetic operations. In the standard arithmetic-operation model, the algorithm performs O(sqrt(N)) iterations and returns the exact integer result. In local Python measurements, QSS was approximately 27.57 to 2406.83 times faster than direct enumeration for the tested sizes N = 10^4 to 10^8. Compared with two alternative sublinear quotient-grouping implementations, the measured advantage was approximately 1.67 to 2.01 times due to reduced constant overhead. These results apply to the specific software implementations and computer configuration tested and do not, by themselves, establish superiority over all existing algorithms.