Extremal and Average Behavior of a Shifted GCD Sum over Divisors

For every integer n >= 2, define A(n) = sum_{d|n} gcd(d-1, n/d-1), and write B(n) = A(n) - 2(n-1). We prove the sharp inequality A(n) <= 9n/4, with equality only at n = 36, and determine the next two extremal ratios: after excluding 36, equality in A(n) <= 20n/9 occurs only at n = 45; after also excluding 45, equality in A(n) <= 31n/14 occurs only at n = 28. The extremal theorem combines an analytic cutoff with an exact finite verification. We establish B(n) = O_epsilon(n^(1/2+epsilon)), show that the exponent 1/2 is optimal on the power scale, classify the integers for which A(n) <= 2n, and give an explicit uniform bound of order n^(3/4). We also prove a three-term average-order formula whose leading term is (3/(2 pi^2)) x (log x)^2, derive quantitative bounds for exceptional sets, construct a Pell-type family with two distinct large gcd contributions, and give an exact summatory formula evaluable in O(sqrt(X) log X) arithmetic operations. This Version 2 substantially extends the original sharp-bound preprint. The uploaded files include the paper, its LaTeX source, and the exact-integer extremal verifier. AI-assistance disclosure: GPT-5.6 Sol generated the original conjecture that A(n) <= 9n/4, with equality only at n = 36. GPT-6 Astra then developed the proof and the subsequent extensions, including the extremal refinements, average-order analysis, Pell-type construction, and exact finite verification. Krystal Hirsch directed the investigation and iterative checking and revision of the mathematics and manuscript, and assumes responsibility for the published work. The full analytic proof remains an unrefereed preprint and has not yet been independently verified by a domain expert.