In this thesis we prove several results in analytic number theory. 1. We show that there exist 3-digit palindromic primes in base b for a set of b having density 1 and that if b is sufficiently large then there is a $3$-digit palindrome in base b having precisely two prime factors. 2. We prove various estimates for averages of sums of Kloosterman fractions over primes. The first of these improves previous results of Fouvry-Shparlinski and Baker. 3. By using the q-analogue of van der Corput's method to estimate short Kloosterman sums we study the divisor function in an arithmetic progression to modulus q. We show that the expected asymptotic formula holds for a larger range of q than was previously known, provided that q has a certain factorisation. 4. Let ‖x‖ denote the distance from x to the nearest integer. We show that for any irrational α and any ϴ< 8/23 there are infinitely many n which are the product of two primes for which ‖nalpha‖ ≤ n ^-ϴ. 5. By establishing an improved level of distribution we study almost-primes of the form f(p,n) where f is an irreducible binary form over Z. 6. We show that for an irreducible cubic f ? Z[x] and a full norm form $mathbf N$ for a number field $K/Q$, satisfying certain hypotheses, the variety $$f(t)=mathbf N(x_1,ldots,x_k) e 0$$ satisfies the Hasse principle. Our proof uses sieve methods.

<p>Given a positive integer k, Conrey, Farmer, Keating, Rubinstein and Snaith conjectured a formula for the asymptotics of the k-th moments of the central values of quadratic Dirichlet L-functions. The conjectured formula for the moments is expressed as sum of a k(k+1)/2 degree polynomial in log |d|. In the sum, d varies over the set of fundamental discriminants. This polynomial, called the moment polynomial, is given as a k-fold residue. In Part I of this thesis, we derive explicit formulae for first k lower order terms of the moment polynomial.</p> <p> In Part II, we present a formula bounding the average of S(t,f), the remainder term in the formula for the number of zeros of an L-function, L(s,f), where f is a newform of weight k and level N. This is Turing's method applied to cuspforms. We carry out the improvements to Turing's original method including using techniques of Booker and Trudgian. These improvements have application to the numerical verification of the Riemann Hypothesis.</p>

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems.

<p>Given a positive integer k, Conrey, Farmer, Keating, Rubinstein and Snaith conjectured a formula for the asymptotics of the k-th moments of the central values of quadratic Dirichlet L-functions. The conjectured formula for the moments is expressed as sum of a k(k+1)/2 degree polynomial in log |d|. In the sum, d varies over the set of fundamental discriminants. This polynomial, called the moment polynomial, is given as a k-fold residue. In Part I of this thesis, we derive explicit formulae for first k lower order terms of the moment polynomial.</p> <p> In Part II, we present a formula bounding the average of S(t,f), the remainder term in the formula for the number of zeros of an L-function, L(s,f), where f is a newform of weight k and level N. This is Turing's method applied to cuspforms. We carry out the improvements to Turing's original method including using techniques of Booker and Trudgian. These improvements have application to the numerical verification of the Riemann Hypothesis.</p>

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems.

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis we consider three different issues of analytic number theory. Firstly, we investigate how residues modulo q may be expressed as products of small primes. In Chapter 1, we work in the regime in which these primes are less than q, and present some partial results towards an open conjecture of Erdös. In Chapter 2, we consider the kinder regime in which these primes are at most q^C , for some constant C that is greater than 1. Here we reach an explicit version of Linnik's Theorem on the least prime in an arithmetic progression, saving that we replace 'prime' with 'product of exactly three primes'. The results of this chapter are joint with Prof. Olivier Ramaré. The next two chapters concern equidistribution modulo 1, specifically the notion that an infinite set of integers is metric poissonian. This strong notion was introduced by Rudnick and Sarnak around twenty years ago, but more recently it has been linked with concepts from additive combinatorics. In Chapter 3 we study the primes in this context, and prove that the primes do not enjoy the metric poissonian property, a theorem which, in passing, improves upon a certain result of Bourgain. In Chapter 4 we continue the investigation further, adapting arguments of Schmidt to demonstrate that certain random sets of integers, which are nearly as dense as the primes, are metric poissonian after all. The major work of this thesis concerns the study of diophantine inequalities. The use of techniques from Fourier analysis to count the number of solutions to such systems, in primes or in other arithmetic sets of interest, is well developed. Our innovation, following suggestions of Wooley and others, is to utilise the additive-combinatorial notion of Gowers norms. In Chapter 5 we adapt methods of Green and Tao to show that, even in an extremely general framework, Gowers norms control the number of solutions weighted by arbitrary bounded functions. We use this result to demonstrate cancellation of the Möbius function over certain irrational patterns.

Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known. Hooley derived a conjectural formula for the average of class numbers of real quadratic fields. In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment, i.e., the average of an arbitrary real power of class numbers. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley's work. In the final chapter we consider the case of the -1 power from a numerical point of view and develop an efficient algorithm to compute the average for the -1 class number power without computing class numbers.

Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known. Hooley derived a conjectural formula for the average of class numbers of real quadratic fields. In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment, i.e., the average of an arbitrary real power of class numbers. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley's work. In the final chapter we consider the case of the -1 power from a numerical point of view and develop an efficient algorithm to compute the average for the -1 class number power without computing class numbers.

Let $a \in \mathbb{Z}$ be a non-zero integer. Let $p$ be a prime such that $p \nmid a$. Define the index of $a$ modulo $p$, denoted $i_{a}(p)$, to be the integer $i_{a}(p) := [(\mathbb{Z}/p\mathbb{Z})^{\ast}:\langle a \bmod{p} \rangle]$. Let $N_{a}(x) := \#\{p \le x:i_{a}(p)=1\}$. In 1927, Emil Artin conjectured that \begin{equation*} N_{a}(x) \sim A(a)\pi(x) \end{equation*} where $A(a)>0$ is a constant dependent only on $a$ and $\pi(x):=\{p \le x: p\text{ prime}\}$. Rewrite $N_{a}(x)$ as follows: \begin{equation*} N_{a}(x) = \sum_{p \le x} f(i_{a}(p)), \end{equation*} where $f:\mathbb{N} \to \mathbb{C}$ with $f(1)=1$ and $f(n)=0$ for all $n \ge 2$.\\ \indent We examine which other functions $f:\mathbb{N} \to \mathbb{C}$ will give us formul\ae \begin{equation*} \sum_{p \le x} f(i_{a}(p)) \sim c_{a}\pi(x), \end{equation*} where $c_{a}$ is a constant dependent only on $a$.\\ \indent Define $\omega(n) := \#\{p|n:p \text{ prime}\}$, $\Omega(n) := \#\{d|n:d \text{ is a prime power}\}$ and $d(n):=\{d|n:d \in \mathbb{N}\}$. We will prove \begin{align*} \sum_{p \le x} (\log(i_{a}(p)))^{\alpha} &= c_{a}\pi(x)+O\left(\frac{x}{(\log x)^{2-\alpha-\varepsilon}}\right) \\ \sum_{p \le x} \omega(i_{a}(p)) &= c_{a}^{\prime}\pi(x)+O\left(\frac{x\log \log x}{(\log x)^{2}}\right) \\ \sum_{p \le x} \Omega(i_{a}(p)) &= c_{a}^{\prime\prime}\pi(x)+O\left(\frac{x\log \log x}{(\log x)^{2}}\right) \end{align*} and \begin{equation*} \sum_{p \le x} d(i_{a}) = c_{a}^{\prime\prime\prime}\pi(x)+O\left(\frac{x}{(\log x)^{2-\varepsilon}}\right) \end{equation*} for all $\varepsilon > 0$.\\ \indent We also extend these results to finitely-generated subgroups of $\mathbb{Q}^{\ast}$ and $E(\mathbb{Q})$ where $E$ is an elliptic curve defined over $\mathbb{Q}$. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-08-03 10:45:47.408