Results in analytic and algebraic number theory

Abstract

The thesis begins with proving some theorems about Gauss sums and Jacobi sums. Using theorems the first chapter ends with a proof that if p is a prime such that p ≡ 1 (mod 4), then there are integers a and b such that p = a2 + b2. In the second chapter some useful results concerning the Dedekind zeta function are proven. Among these results are that the Dedekind zeta function is meromorphic with a simple pole at s = 1. The third chapter has a new result concerning Carmicheal numbers. Specifically let α, β ∈ ℝ be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence Bα,β = (⌊αn = β⌋)∞n=1. The chapter concludes with heuristic evidence via Dickson's conjecture to support our conjecture that we obtain same result when is an irrational number of infinite type. In the fourth chapter we show that for any finite Galois extension K of the rational numbers Q, there are infinitely many Carmichael numbers composed solely of primes for which the associated class of Frobenius automorphisms coincides with any given conjugacy class of Gal(K∣ℚ). The result has three corollaries: for any algebraic number field K, there are infinitely many Carmichael numbers which are composed solely of primes that split completely in K; for every natural number n, there are infinitely many Carmichael numbers of the form a2 +nb2 with a, b integers; and there are infinitely many Carmichael numbers composed solely of primes p ≡ a (mod d) with a, d coprime. Finally, in chapter five we prove a new result regarding Piatetski-Shapiro primes in relation to almost primes. We show that for any fixed c ∈ (1, 77/ 76) there are infinitely many primes of the form p = ⌊nc⌋, where n is a natural number with at most eight prime factors (counted with multiplicity).