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Various problems on subproducts of residue classes modulo a prime Parvardi, Amir Hossein


Let p be a prime number. Booker and Pomerance find an integer y with 1 < y ≤ p such that all non-zero residue classes modulo p can be written as a square-free product of positive integers up to y. Let us denote by y(p) the smallest such y. Booker and Pomerance show in their paper that except for p - 5 and 7, we have y(p) ≤ y and some better upper bounds were conjectured. Later, Munsch and Shparlinski proved those conjectures with even better localization. Their work was done as the same time as ours, but with fairly more complicated methods in the proof. We were seeking to find a solution for the problem using Pólya-Vinogradov inequality or at most its improvement, the Burgess bound on character sums. That being said, we removed the condition in the problem that the product has to be square-free. We proved that for m > 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).

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