Selberg sieve lower bound for S in dimension 2: hyperbola plus lcm fibres, no integrals

For the twin sieve (nu(d)=tau(d)/d, odd primes), getting S >= (log z)^2/8 in Lean went in three steps. (1) Group pairs (r,s) of odd squarefree numbers with rs <= z by l = lcm(r,s). Inject (r,s) into the sigma (l, r, t = gcd(r,s)); s is recovered as (l/r)*gcd, and k(s) = k(l/r)k(gcd) by coprimality, since l is squarefree. The fibre sum is then the Dirichlet product (k.pmul (zeta*k)) * k at l, which is multiplicative, so you compare it with g(l) prime by prime: 2/(p-1) + 1/(p-1)^2 <= 2/(p-2). (2) Replace 1/n by k(rad n) with the existing radical injection, once for each variable, swapping the sums with sum_comm' and the iff s <= floor(z/a) <-> a*s <= z (Nat.le_floor_iff' then le_div_iff0). (3) The analytic bounds sum_{n<=y odd} 1/n >= (log y)/2 and sum (log y - log n)/n >= (log y)^2/4 come from telescoping u_i = min(log(2i+1), log y) with Finset.sum_range_sub, using 1/a >= (u_{i+1}-u_i)/2 and (b-a)(2L-a-b) <= 4c(L-a); nlinarith closes the step. This beat the square a,b <= sqrt z with (a,b)=1 by a factor 2.5.