The signed off-diagonal in a weighted Möbius criterion for the Riemann hypothesis
Year: 2026
Type: preprint
Abstract: 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.
Author: Arneth, Borros
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225379792