Adleman–Pomerance–Rumely conjecture on the maximal order constant for ω*(n)

Establish that there exist infinitely many integers n such that ω*(n) > exp(((log 2 + o(1))·log n)/(log log n)), where ω*(n) denotes the number of primes p for which p − 1 divides n. This upgrades the constant 1/2·log 2 appearing in Prachar’s GRH-conditional lower bound to log 2 and would be best possible in view of the maximal order of the divisor function.

Background

The shifted-prime divisor function ω*(n) counts primes p with p−1 dividing n. Prachar obtained a GRH-conditional bound with constant (1/2)·log 2 in the exponent, while Adleman–Pomerance–Rumely (APR) established the same shape unconditionally with some unspecified constant. The conjecture seeks the optimal constant log 2 in the exponent for infinitely many n, matching the known upper bound for the divisor function τ(n).

Achieving log 2 would settle the maximal order question in the strongest possible form and has implications for the Carmichael function λ(n), connecting multiplicative group exponents to the distribution of shifted primes.