OAI.TotientAsymptotic.coefficient_nonnegative

The theorem states that, for every natural number H and every real s in the half-open interval [0,1), the quantity AH(H, 1, s) is nonnegative, where 1 denotes the constant function one. Here AH(H,f,s) is the H-th approximating coefficient in the totient asymptotics: it equals ρ^(H(H−1)/2) · (γ/α(s))^H times a sum (finsum) over natural numbers d that are totient values (d = φ(n) for some n > 0), of f(ℓ(d)/d)/d, with ℓ(d) the least positive n having φ(n) = d, multiplied by an inner finsum over nonempty finite sets T of tail data contained in the set of witnesses for (H, s, d). Each such term is (−1)^(|T|−1) · exp(−(γ/α(s)) · Σ_{n≥0} ρ^(H+n) · maxD(H+n, T)), where maxD is the maximum over T of a weighted sum D of logarithms of logarithms of the tail primes. Here ρ is the infimum of the set of z in (0,1) with Σ_{j≥1} a(j) z^j = 1, γ is the reciprocal of Σ_{j≥1} j·a(j)·ρ^j, a(j) = (j+1)log(j+1) − j log j − 1, and α(s) = λ e^{λs} with λ = log(1/ρ). With f constant 1, the claim is that this alternating inclusion-exclusion expression is never negative.