Get started
Work

A single fixed-endpoint Möbius input for the Goldbach reduction

Year: 2026
Type: preprint
Abstract: A fixed-endpoint, fixed-residue Bombieri–Vinogradov-type estimate for Λ(n)μ(N−n) over squarefree moduli up to √N (log N)^12, with unweighted saving (log N)^(−3), implies ≫ N/(log N)^3 unordered Goldbach prime pairs. Ordinary Bombieri–Vinogradov supplies the untwisted input. Full divisor cancellation switches complementary sums to ordinary small-divisor sums and gives arbitrary logarithmic saving for the signed μ and μ log discrepancies at fixed power cutoffs above √N. The cancellation, residual relation, and a fixed-endpoint absolute-value sufficient condition at a power cutoff occur in Lee's manuscript (September 2026). This independently prepared note specifies the logarithmic square-root cutoff and unweighted saving; every fixed exponent greater than 11 suffices. The signed cancellation bypasses the separate μ log input in Cantarini's fixed-endpoint formulation in this range. Huang–Li already derive the logarithmically weighted estimate from their endpoint-uniform twisted hypothesis; here the remaining input is required only at one endpoint. We give a direct remainder identity, weaker signed sufficient conditions, ordinary evaluations of the moving tail and corrected triangular error, and a general ordinary-level formulation. Davenport's estimate and Parseval yield an unconditional mean-square bound for both weights and all coprime moduli. For fixed 0 < θ < 1 and A, H > 0, the two bounds of Cantarini's Conjecture 6 at level N^θ hold simultaneously outside O(X/(log X)^H) totals in [X, 2X]. This recovers a classical almost-all Goldbach consequence. The input for every sufficiently large even N remains an assumption, so the pointwise Goldbach conclusion is conditional. Revision 11 (5 November 2026). Comments are welcome.
DOI: 10.5281/zenodo.24093999
Language: English
Open Access status: green
OpenAlex ID: W7225637684
Cites:
Cited by:
Related to: