Maier 1984 third iterates phi sigma functions

../

---

H. Maier, On the third iterates of the phi- and sigma-functions. Colloquium Mathematicum 49 (1984), 123-130.

Maier studies the counting functions N_sigma(k,alpha,x), the number of n <= x with sigma_k(n) < alpha n, and N_phi(k,alpha,x), the number of n <= x with phi_k(n) > alpha n, extending earlier upper bounds of Erdos to lower bounds for k = 3. Theorem 1 shows N_sigma(3,alpha,x) > x/log^2 x for alpha > alpha_0 and x large, which yields in particular liminf sigma_3(n)/n < infinity, an unconditional proof of Schinzel's conjecture for k = 3 (previously known only for k = 1, 2, and for general k only under Schinzel's Hypothesis H via Makowski). Theorem 2 is the analogous lower bound N_phi(3,alpha,x) > x/log^2 x for alpha < alpha_1, and the paper indicates the refinements (4) and (5) that insert an extra factor (log log x)^t for arbitrarily large t. The method builds a set of n whose prime factorization is engineered so that the sigma or phi cascade stays controlled, using the auxiliary quantities rho(n) = sum_{p | n} 1/p and lambda(p) = sigma(p+1)/(p+1). Problem 410 asks whether sigma_k(n)^{1/k} tends to infinity as k grows, for each fixed n >= 2. Maier's theorems concern the single iterate k = 3 and how small sigma_3(n)/n can be as n varies, so they are background on iterated sigma for that question and say nothing about growth in k.

Source: retained PDF. No notice is printed in the file; the publisher's article record offers the PDF "Free download under CC-BY license" ("Pobierz zgodnie z CC-BY" as the Polish page prints it), no version named (https://www.impan.pl/get/doi/10.4064/cm-49-1-123-130, read 2026-10-02): the Creative Commons Attribution license with no version named.

Bears on. #410

Results to transcribe.

• Theorem 1: For alpha > alpha_0 and x > x_0(alpha), N_sigma(3,alpha,x) > x/log^2 x; in particular liminf_{n} sigma_3(n)/n < infinity, proving Schinzel's conjecture for k = 3.
• Theorem 2: For alpha < alpha_1 and x > x_1(alpha), N_phi(3,alpha,x) > x/log^2 x.
• Refinements (4) and (5): The bounds improve to N_sigma(3,alpha,x) > (x/log^2 x)(log log x)^t and N_phi(3,alpha,x) > (x/log^2 x)(log log x)^t for arbitrarily large t.