Lola Thompson

Bounded gaps between primes in number fields and function fields

The Hardy--Littlewood prime

Counting and effective rigidity in algebra and geometry

k

Arithmetic functions at consecutive shifted primes

-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field

VARIATIONS ON A THEOREM OF DAVENPORT CONCERNING ABUNDANT NUMBERS

\mathbb{F}_q(t)

ON THE DIVISORS OF xn – 1 IN 𝔽p[x]

.

On integers

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.

n

For each of the functions f ∈ {φ, σ, ω, τ} and every natural number K, we show that there are infinitely many solutions to the inequalities f(pn - 1) f(pn+1 - 1) > ⋯ > f(pn+K -1). We also answer some questions of Sierpinski on the digit sums of consecutive primes. The arguments make essential use of Maynard and Taos method for producing many primes in intervals of bounded length.

for which

DOI: Let \(\sigma(n)\) denote the sum of the positive divisors of \(N\). In 1933, Davenport showed that \(n/\sigma(n)\) possesses a continuous distribution function. We study the behavior of analogous weighted distributions involving certain complex-valued multiplicative functions. Our results cover many of the more frequently encountered functions, including \(\sigma(n)\), \(\tau(n)\) and \(\mu(n)\). They also apply to the representation function for sums of two squares, leading again to a continuous distribution function. 10.1017/S0004972713000695

X^n-1

In a recent paper, we considered integers n for which the polynomial xn – 1 has a divisor in ℤ[x] of every degree up to n, and we gave upper and lower bounds for their distribution. In this paper, we consider those n for which the polynomial xn – 1 has a divisor in 𝔽p[x] of every degree up to n, where p is a rational prime. Assuming the validity of the Generalized Riemann Hypothesis, we show that such integers n have asymptotic density 0.

has a divisor of every degree

A positive integer