Refine and Prove the Congruence Monoid Prime Count Estimate

Establish a corrected asymptotic estimate for T_d(x), the number of congruence monoid primes no greater than x in A_d = {n in N : n ≡ 1 (mod d)}, that better matches empirical data than the proposed estimate x/(d (ln x)^{1/d}); then prove the correctness of the refined estimate rigorously.

Background

Section 2 defines the congruence monoid A_d = {n in N : n ≡ 1 (mod d)} and congruence monoid primes. The authors propose the heuristic estimate T_d(x) ≈ x/(d (ln x)^{1/d}) for the number of such primes up to x, and present empirical evidence indicating the estimate is close but systematically deviates beyond certain x-ranges.

Open Problem 2.5 asks for improving this conjectural estimate to better fit the data and to provide a proof, analogous in spirit to the classical Prime Number Theorem but within the congruence monoid setting.