OAI.TotientAsymptotic.weighted_totient_one_two

The theorem states that, for each k in {1, 2}, there is a real constant c > 0 such that c ≤ A(f_k)(s) for every s in the half-open interval [0,1). Here f_k(r) = min(1,(k+1)/r) − min(1,k/r) is the weight attached to the ratio r = ℓ(d)/d, where ℓ(d) is the least positive n with φ(n) = d (and 0 if d is not a totient value). A(f)(s) is defined as the limit as H tends to infinity of the quantity AH(H,f,s), a ρ-, γ- and α(s)-normalized alternating inclusion-exclusion sum over finite sets of tail data (tuples of primes and a cofactor satisfying the witness conditions for s) that is summed over totient values d with f(ℓ(d)/d)/d weights, where ρ is the smallest root in (0,1) of the renewal equation built from the coefficients a(j). Thus the statement is a uniform positive lower bound on A(f_1) and A(f_2) over s in [0,1).