arXiv subjects

Ayla Gafni

Publications and source records attributed to Ayla Gafni.

11 recordsLinked to original sources

arXiv2026

Exponential sums weighted by additive functions

We introduce a general class $F_0$ of additive functions $f$ such that $f(p) = 1$ and prove a tight bound for exponential sums of the form $\sum_{n \le x} f(n) e(αn)$ where $f \in F_0$ and $e(θ) = \exp(2πi θ)$. Both $ω$, the number of distinct primes of $n$, and $Ω$, the total number primes of $n$, are members of $F_0$. As an application of the exponential sum result, we use the Hardy-Littlewood circle method to find the asymptotics of the Goldbach-Vinogradov ternary problem associated to $Ω$, namely we show the behavior of $r_Ω(N) = \sum_{n_1+n_2+n_3=N}Ω(n_1)Ω(n_2)Ω(n_3)$, as $N \to \infty$. Lastly, we end with a discussion of further applications of the main result.

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arXiv2025

Rough numbers between consecutive primes

Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap, confirming a prediction of Erdős. In fact the number $N(X)$ of exceptional gaps with $p_n \in [X,2X]$ is shown to be at most $O(X/\log^2 X)$. Assuming a form of the Hardy--Littlewood prime tuples conjecture, we establish a more precise asymptotic $N(X) \sim c X / \log^2 X$ for an explicit constant $c>0$, which we believe to be between $2.7$ and $2.8$. To obtain our results in their full strength we rely on the asymptotics for singular series developed by Montgomery and Soundararajan.

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arXiv2025

On the number of exceptional intervals to the prime number theorem in short intervals

For a fixed exponent $0 < θ\leq 1$, it is expected that we have the prime number theorem in short intervals $\sum_{x \leq n < x+x^θ} Λ(n) \sim x^θ$ as $x \to \infty$. From the recent zero density estimates of Guth and Maynard, this result is known for all $x$ for $θ> \frac{17}{30}$ and for almost all $x$ for $θ> \frac{2}{15}$. Prior to this work, Bazzanella and Perelli obtained some upper bounds on the size of the exceptional set where the prime number theorem in short intervals fails. We give an explicit relation between zero density estimates and exceptional set bounds, allowing for the most recent zero density estimates to be directly applied to give upper bounds on the exceptional set via a small amount of computer assistance.

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arXiv2024

Improved bounds on number fields of small degree

We study the number of degree $n$ number fields with discriminant bounded by $X$. In this article, we improve an upper bound due to Schmidt on the number of such fields that was previously the best known upper bound for $6 \leq n \leq 94$.

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arXiv2021

Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree $n$ polynomials $f$ with $\mathrm{Gal}(f) \subseteq A_n$. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree $n$ monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree $n$ number fields with almost prime discriminants.

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arXiv2021

Almost all primes satisfy the Atkin-Serre conjecture and are not extremal

Let $f(z)=\sum_{n=1}^{\infty} a_f(n)e^{2πi n z}$ be a non-CM holomorphic cupsidal newform of trivial nebentypus and even integral level $k\geq 2$. Deligne's proof of the Weil conjectures shows that $|a_f(p)|\leq 2p^{\frac{k-1}{2}}$ for all primes $p$. We prove for 100% of primes $p$ that $2p^{\frac{k-1}{2}}\frac{\log\log p}{\sqrt{\log p}}<|a_f(p)|<\lfloor 2p^{\frac{k-1}{2}}\rfloor$. Our proof gives an effective upper bound for the size of the exceptional set. The lower bound shows that the Atkin-Serre conjecture is satisfied for 100% of primes, and the upper bound shows that $|a_f(p)|$ is as large as possible (i.e., $p$ is extremal for $f$) for 0% of primes. Our proofs use the effective form of the Sato-Tate conjecture proved by the second author, which relies on the recent proof of the automorphy of the symmetric powers of $f$ due to Newton and Thorne.

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arXiv2021

Partitions into prime powers

For a subset $\mathcal A\subset \mathbb N$, let $p_{\mathcal A}(n)$ denote the restricted partition function which counts partitions of $n$ with all parts lying in $\mathcal A$. In this paper, we use a variation of the Hardy-Littlewood circle method to provide an asymptotic formula for $p_{\mathcal A}(n)$, where $\mathcal A$ is the set of $k$-th powers of primes (for fixed $k$). This combines Vaughan's work on partitions into primes with the author's previous result about partitions into $k$-th powers. This new asymptotic formula is an extension of a pattern indicated by several results about restricted partition functions over the past few years. Comparing these results side-by-side, we discuss a general strategy by which one could analyze $p_{\mathcal A}(n )$ for a given set $\mathcal A$.

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arXiv2018

Additive energy and the metric Poissonian property

Let $A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of $A$ (the number of solutions to $a_1 + a_2 = a_3 + a_4$ with $a_i \in A$) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of $A$ modulo $1$. There appears to be reasonable evidence to speculate a sharp Khintchine-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.

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arXiv2015

Power Partitions

In 1918, Hardy and Ramanujan published a seminal paper which included an asymptotic formula for the partition function. In their paper, they also claim without proof an asymptotic equivalence for $p^k(n)$, the number of partitions of a number $n$ into $k$-th powers. In this paper, we provide an asymptotic formula for $p^k(n)$, using the Hardy-Littlewood Circle Method. We also provide a formula for the difference function $p^k(n+1)-p^k(n)$. As a necessary step in the proof, we obtain a non-trivial bound on exponential sums of the form $\sum_{m=1}^q e(\frac{am^k}{q})$.

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arXiv2014

Counting rational points near planar curves

We find an asymptotic formula for the number of rational points near planar curves. More precisely, if $f:\mathbb{R}\rightarrow\mathbb{R}$ is a sufficiently smooth function defined on the interval $[η,ξ]$, then the number of rational points with denominator no larger than $Q$ that lie within a $δ$-neighborhood of the graph of $f$ is shown to be asymptotically equivalent to $(ξ-η)δQ^2$.

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arXiv2009

Longest Run of Equal Parts in a Random Integer Composition

This note examines a problem in enumerative and asymptotic combinatorics involving the classical structure of integer compositions. What is sought is an analysis on average and in distribution of the length of the longest run of consecutive equal parts in a composition of size n. The problem was recently posed by Herbert Wilf (see arXiv: 0906.5196).

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