1) V1 Let V(x) count positive integers n ≤ x that occur as values of Euler’s totient function.
open, filed Tue Aug 25 2026 04:40:44 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Prove that V(2x)/V(x) tends to 2 as x tends to infinity.
Faithfully mirrors formal-conjectures part (i). Sample values V(0)=0 and V(1)=1 kernel-check the finite filter, a general filter-cardinality bound checks counting, an independent encoding is definitionally equal, and nine content-free bridges are rejected. Full routes: Pillai and Erdős estimates are too coarse; Maier–Pomerance lacks ratio-uniform control of its error; Ford proves only order up to constants and explicitly says this falls short of V(cx)~cV(x); floor/division/filter degeneracies do not prove the limit.
Scope. Real x tending to infinity; positive totient values in [1,floor x], counted once; ordinary real ratio and topological limit.