Identifying ε: Möbius Structure and the Composite Variance in the Gauss Circle Problem

Hardy conjectured that the classical Gauss circle error satises E (x) = O (x1/4+ε) for every ε>0. This paper analyses a related quantity: the uctuation ER (x) = N (x) −x/ (2π) in the primitive coprime count N (x) at a single scale. We show that ER admits an exact Möbius decomposition into 2ω (s) fractional-part terms, where ω (s) is the number of distinct prime factors of s= m+ n. For prime s, this reduces to two terms with constant variance. For composite s, we identify the per-s variance formula Var (e|s) = A (s) /12*φ (s) /s+Var (T) /4, A (s) = 2^ω (s) +1−Prod p|s (1 +1/p²), empirically, with no free parameters; a rigorous derivation is open. By the HardyRamanujan theorem (ω (s) ∼log log s), this gives σc ∼ (ln s) ^ (ln 2) /2, conrmed numerically to 1/4 and remains open below.