Generalised divisor sums of binary forms over number fields

Estimating averages of Dirichlet convolutions $1 \ast \chi$, for some real Dirichlet character $\chi$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin's conjecture for Ch\^atelet surfaces. We introduce a far-reaching generalization of this problem, in particular replacing $\chi$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin's conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number

Short Character Sums and Their Applications

In analytic number theory, the most natural generalisations of the famous Riemann zeta function are the Dirichlet L-functions. Each Dirichlet L-function is attached to a q-periodic arithmetic function for some natural number q, known as a Dirichlet character modulo q. Dirichlet characters and L-functions encode with them information about the primes, especially in reference to their remainders modulo q. For this reason, number theorists are often interested in bounds on short sums of a Dirichlet character over the integers. For one, such sums appear as an intermediate step in partial summation bounds for Dirichlet L-functions. However, short character sum estimates may also be used directly to tackle other number theoretic problems. In this thesis, we wish to examine the application of character sum estimates in some specific settings. There are three main estimates that we are interested in: the trivial bound, Burgess’ bound, and the Pólya–Vinogradov inequality. Of these, we will focus primarily on Burgess’ bound; the main result of this thesis will be the computation of explicit constants versions of Burgess’ bound for a variety of parameters, improving upon the work of Treviño [54]. The interplay between Burgess’ bound and the Pólya–Vinogradov inequality is vital in this explicit setting, and we will dedicate a portion of this thesis to investigating these interactions. This makes precise the work of Fromm and Goldmakher [21], who demonstrated a counterintuitive influence the Pólya–Vinogradov inequality has on Burgess’ bound. Once we have established improvements to Burgess’ bound in the explicit setting, we will “test” these improvements by tackling several applications where Burgess’ bound has been used previously. Primary among these is an explicit bound for L-functions across the critical strip. We also include applications to norm-Euclidean cyclic fields and least kth power non-residues

The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture

For each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that n(p) = O(p^\eps) for every fixed \eps>0. This conjecture follows from the generalised Riemann hypothesis, and is known to hold for almost all primes $p$ but remains open in general. In this paper we show that Vinogradov's conjecture also follows from the Elliott-Halberstam conjecture on the distribution of primes in arithmetic progressions, thus providing a potential "non-multiplicative" route to the Vinogradov conjecture. We also give a variant of this argument that obtains bounds on short centred character sums from "Type II" estimates of the type introduced recently by Zhang and improved upon by the Polymath project, or from bounds on the level of distribution on variants of the higher order divisor function. In particular, we can obtain an improvement over the Burgess bound would be obtained if one had Type II estimates with level of distribution above $2/3$ (when the conductor is not cube-free) or $3/4$ (if the conductor is cube-free); morally, one would also obtain such a gain if one had distributional estimates on the third or fourth divisor functions $\tau_3, \tau_4$ at level above $2/3$ or $3/4$ respectively. Some applications to the least primitive root are also given.Comment: 25 pages, no figures. A variant of the argument using distribution estimates of divisor-type functions, which gives somewhat better numerical exponents, has been added; also some new references added, and referee suggestions incorporate

Double Character Sums over Subgroups and Intervals

We estimate double sums $$S_\chi(a, I, G) = \sum_{x \in I} \sum_{\lambda \in G} \chi(x + a\lambda), \qquad 1\le a < p-1,$$ with a multiplicative character $\chi$ modulo $p$ where $I= \{1,\ldots, H\}$ and $G$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_\chi(a, I, G)$ can be derived from the Burgess bound if $H \ge p^{1/4+\varepsilon}$ and from some standard elementary arguments if $T \ge p^{1/2+\varepsilon}$, where $\varepsilon>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums $$T_\chi(a, G) = \sum_{\lambda, \mu \in G} \chi(a + \lambda + \mu), \qquad 1\le a < p-1,$$ and give an application to primitive roots modulo $p$ with $3$ non-zero binary digits