Let E be a one-parameter family of elliptic curves over a number field. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any non-degenerate family E has average root number 0, provided that two classical arithmetical conjectures hold for two homogeneous polynomials with integral coefficients constructed explicitly in terms of E.<br />The first such conjecture -- commonly associated with Chowla -- asserts the equidistribution of the parity of the number of primes dividing the integers represented by a polynomial. We prove the conjecture for homogeneous polynomials of degree 3.<br />The second conjecture used states that any non-constant homogeneous polynomial yields to a square-free sieve. We sharpen the existing bounds on the known cases by a sieve refinement and a new approach combining height functions, sphere packings and sieve methods.

Circular words, also known as necklaces, are combinatorial objects closely related to linear words. A brief history of circular words is given, from their early conception to present results. We introduce the concept of a level word, that being a word containing a equal or roughly equal amount of each letter. We characterize exactly the lengths for which level square free circular words on three letters exist. This is accomplished through a modification of Shur’s construction of square-free circular words. A word on two letters is called a Frankel-Simpson word if the only squares it contains are 00, 11, and 0101. Using the result mentioned above and several computer searches, we characterize exactly the lengths for which circular Frankel-Simpson words exist, and give an example or construction for each. / Graduate

The thesis is structured into 5 chapters as follows: <strong>Chapter 1</strong> is an introduction to the tools and methods we use most frequently. <strong>Chapter 2</strong> Pairs of k-free Numbers, consecutive square-full Numbers. In this chapter, we refine the approximate determinant method by Heath-Brown. We present applications to asymptotic formulas for consecutive k-free integers, and more generally for k-free integers represented by r-tuples of linear forms. We also show how the method can be used to derive an upper bound for the number of consecutive square-full integers. Finally, we apply the method to make a statement about the size of the fundamental solution of Pell equations. <strong>Chapter 3</strong> Power-Free Values of Polynomials. A conjecture by Erdös states that for any irreducible polynomial f of degree d&ge;3 with no fixed (d-1)-th power prime divisor, there are infinfinitely many primes p such that f(p) is (d-1)-free. We prove this conjecture and derive the corresponding asymptotic formulas. <strong>Chapter 4</strong> Integer Points on Bilinear and Trilinear Equations. In the fourth chapter, we derive upper bounds for the number of integer solutions on bilinear or trilinear forms. <strong>Chapter 5</strong> In the fifth chapter, we present a method to count the monomials that occur in the projective determinant method when the method is applied to cubic varieties.

Our research is in the general area of String Algorithms and Combinatorics on Words. Specifically, we study a generalization of square-free strings, shuffle properties of strings, and formalizing the reasoning about finite strings. The existence of infinitely long square-free strings (strings with no adjacent repeating word blocks) over a three (or more) letter finite set (referred to as Alphabet) is a well-established result. A natural generalization of this problem is that only subsets of the alphabet with predefined cardinality are available, while selecting symbols of the square-free string. This problem has been studied by several authors, and the lowest possible bound on the cardinality of the subset given is four. The problem remains open for subset size three and we investigate this question. We show that square-free strings exist in several specialized cases of the problem and propose approaches to solve the problem, ranging from patterns in strings to Proof Complexity. We also study the shuffle property (analogous to shuffling a deck of cards labeled with symbols) of strings, and explore the relationship between string shuffle and graphs, and show that large classes of graphs can be represented with special type of strings. Finally, we propose a theory of strings, that formalizes the reasoning about finite strings. By engaging in this line of research, we hope to bring the richness of the advanced field of Proof Complexity to Stringology. / Thesis / Doctor of Philosophy (PhD)