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Counting integers with restrictions on their prime factors

University of British Columbia

In this thesis, we examine two problems that, on the surface, seem like pure group theory problems, but turn out to both be problems concerning counting integers with restrictions on their prime factors. Fixing an odd prime number q and a finite abelian q-group H=ℤqᵅ₁×ℤqᵅ₂×⋯×ℤqᵅʲ, our first aim is to find a counting function, D(H,x), for the number of integers n up to x such that H is the Sylow q-subgroup of (ℤ/nℤ)×. In Chapter 2, we prove that D(H,x)∼ K_H x(log log x)ʲ/(log x)⁻¹/⁽q⁻¹⁾, where K_H is a constant depending on H. The second problem that we examine in this thesis concerns counting the number of n up to x for which (ℤ/nℤ)× is cyclic and for which (ℤ/nℤ)× is maximally non-cyclic, where (ℤ/nℤ)× is said to be maximally non-cyclic if each of its invariant factors is squarefree. In Chapter 3, we prove that the number of n up to x such that (ℤ/nℤ)× is cyclic is asymptotic to (3/2)x/log x and that the number of n up to x such that (ℤ/nℤ)× is maximally non-cyclic is asymptotic to C_f x/(log x)¹⁻ξ, where ξ is Artin's constant and C_f is the convergent product, C_f=(15/14Γ(ξ)) limₓ→∞ (∏_p≤ₓ\\{p-₁ square-free} (1+(1/p)+(1/p²)) ∏_p≤ₓ (1-(1/p))^ξ). It turns out that both of these problems can be reduced to problems of counting integers with restrictions on their prime factors. This allows the problems to be addressed by classical techniques of analytic number theory.

Text

2019-06-06

Attribution-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70561

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nd/4.0/