Various problems on subproducts of residue classes modulo a prime
Amir Parvardi · cIRcle (University of British Columbia) · 2019
Let p be a prime number. Booker and Pomerance find an integer y with 1 p^[1/(4√e+o(1)], each residue class b of (ℤ/pℤ)^× can be written as a product of elements of the set {1, 2,..., m} modulo p. In fact, we showed that the number of such sub-products (congruent to b mod p) is asymptotic to 2^m/(p - 1). The proofs are based on an identity involving sums of Dirichlet characters modulo p as well as the Burgess inequality on partial character sums. Basically, we use proof by contradiction to state that if the error term (for number of sub-product) is large, then there should be many χ values close to 1, which would result in the character sum being large, thereby contradicting the Burgess inequality (which essentially says bounds the character sums).