Levels p ² q for distinct primes p≠2 and q

Abstract

In six sections we will discuss eta products of weight 1 with levels N=p ² q where p,q are distinct primes. For these levels the number of positive divisors is σ ₀(N)=6. In the present section we treat the case that p is odd, and here most of the effort is needed for level N=18. Five more sections will be dedicated to the case p=2, that is, levels N=4q with odd primes q. We need so much space for the results in the case p=2 due to the fact known from Theorem 3.9, part (3), that the number of holomorphic eta products of a given weight increases when a prime in the factorization of the level is replaced by a smaller prime.