Gregory Martin

Finding reasonably good upper and lower bounds for the clique number of Paley graphs and generalized Paley graphs is an old and open problem in additive combinatorics. In this thesis, we use polynomial methods, together with various tools from number theory, graph theory, and combinatorics, to study this problem. Specifically, we obtain improved upper bounds on the clique number of Paley graphs and generalized Paley graphs over a finite field. We also obtain new upper bounds on the number of distinct roots of lacunary polynomials and improve lower bounds on the number of directions determined by a Cartesian product in an affine Galois plane over a finite field.

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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.

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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).

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We study the limit points Q' of a three-dimensional set Q which encodes the reciprocal qualityof abc triples as the components of a vector of the form (log Rad a, log Rad b, log Rad c) / log c. We establish that if the abc Conjecture holds, Q' is contained in a heptahedron. Unconditionally, we establish the existence of a subset of Q' with non-zero measure. We determine the implications of previous research on related problems involving limit points of abc triples in one-dimensional sets on Q' and discuss possible avenues for future study.

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The class number problem is one of the central open problems of algebraic number theory. It has long been known that there are only finitely many imaginary quadratic fields of class number one, and the full list of such fields is given by the Stark-Heegner theorem, but the corresponding problem for real quadratic fields is still open. It is conjectured that infinitely many real quadratic fields have class number one but at present it is still unknown even whether infinitely many algebraic number fields have class number one. This thesis reviews the relevant work that has been done on this problem in the last several decades. It is primarily concerned with a heuristic model of the sequence of real quadratic class groups called the Cohen-Lenstra heuristics, since this appears to offer the best hope of potentially solving the class number problem. The work of several other people who have put forward interpretations of the Cohen-Lenstra heuristics in other contexts, or who have generalized the heuristics, is also reviewed.

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In this thesis we extend two important theorems in analytic prime number theory to a the setting of Beurling primes, namely The Erdős–Kac theorem and a theorem of Sathe and Selberg. The Erdős–Kac theoremasserts that the number of prime factors that divide an integer n is, in some sense, normally distributed with mean log log n and variancelog log n. Sathe proved and Selberg substantially refined a formula for the counting function of products of k primes with some uniformity on k. A set of Beurling primes is any countable multiset of the reals with elements that tend towards infinity. The set of Beurling primes has a corresponding multiset of Beurling integers formed by all finite products of Beurling primes. We assume that the Beurling integer counting function is approximately linear with varying conditions on the error term in order to prove the stated results. An interesting example of a set of Beurling primes is the set of norms of prime ideals of the ring of integers of a number field. Recently, Granville and Soundararajan have developed a particularly simple proof of the Erdős–Kac theorem which we follow in this thesis. For extending the theorem of Selberg and Sathe much more analytic machinery is needed.

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