The signed off-diagonal in a weighted Möbius criterion for the Riemann hypothesis

We study the summatory Möbius function restricted to integers coprime to six and weighted by the product of the factors over their prime divisors. A finite convolution identity expresses its value at a sharp endpoint through a symmetric double sum with both indices at most the square root of the endpoint. We compute the diagonal as , where is an explicit positive Euler product. The prime number theorem then gives the full off-diagonal as unconditionally. We prove that an error of order in this asymptotic, for every positive , is equivalent to the Riemann hypothesis. A further decomposition bounds the inverse-coefficient counting error on the small-product range at the required scale. On the complementary range the signed error must be controlled jointly with a truncated Euler sum. This last estimate remains open.