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Thomas Wright

Publications and source records attributed to Thomas Wright.

17 recordsLinked to original sources
arXiv2026

$(a,a)$-Carmichael numbers and greatest common divisors of $p-a$

Define an $(a,a)$-Carmichael number to be a squarefree natural number $n$ such that $p\mid n$ implies $p-a\mid n-a$. For such a number $n$ with prime factors $p_1,\cdots,p_m$, define $$K=GCD[p_1-a,\cdots,p_m-a],$$ and let $C_ν(X,a)$ denote the number of $(a,a)$-Carmichael numbers up to $X$ such that $K=ν$. Assuming a strong conjecture on the first prime in an arithmetic progression, we prove that for any integer $a$ and for any natural number $ν$ with $(ν,a)=1$ and $a$ and $ν$ having opposite parity, $$C_ν(X,a)\geq X^{1-(2+o(1))\frac{\log\log\log \log X}{\log\log\log X}}.$$ This is a departure from many traditional constructions of Carmichael numbers, which generally require $K$ to grow along with $n$.

math.NT↗
arXiv2026

Trilinear Kloosterman fractions II: subdyadic intervals and nearly balanced convolutions

This paper broadens the range on which Fouvry and Radziwiłł's results on nearly balanced convolutions apply. In particular, let $α_m$ and $β_n$ be sequences supported on $m\sim M$ and $n\sim N$ where $β_n$ is equidistributed for small moduli, and let $Q=X^{\frac 12+\varepsilon}$. We find that \begin{gather*}\sum_{q\sim Q}\left|\mathop{\sum\sum}_{\substack{n\sim N,m\sim M \\ mn\equiv a\pmod q}}α_mβ_n-\frac{1}{ϕ(q)}\mathop{\sum\sum}_{\substack{n\sim N,m\sim M \\ (mn,q)=1}}α_mβ_n\right|\ll \frac{X}{\log^A X} \end{gather*} if $N=X^{\frac 12+δ}$ and $M=X^{\frac 12-δ}$ with $0<δ<\frac 1{68}$, which improves Fouvry and Radziwiłł's $0<δ<\frac 1{112}$. To prove this, we sharpen Bettin and Chandee's famous result on trilinear forms with Kloosterman fractions in the case where some of the sums are over subdyadic intervals.

math.NT↗
arXiv2026

Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions

In this paper, we improve on Fouvry and Radziwiłł's results on unbalanced convolutions. In particular, we find that if $(α_m)$ and $(β_n)$ are sequences supported on $m\sim M$ and $n\sim M$ where $β_n$ is equidistributed for small moduli, then \begin{gather*}\sum_{q\sim Q}\left|\mathop{\sum\sum}_{\substack{n\sim N,m\sim M \\ mn\equiv a\pmod q}}α_mβ_n-\frac{1}{ϕ(q)}\mathop{\sum\sum}_{\substack{n\sim N,m\sim M \\ (mn,q)=1}}α_mβ_n\right|\ll \frac{X}{\log^A X}, \end{gather*} as long as $\exp((\log x)^{\varepsilon}) \leq N \leq Q^{-11/12} X^{17/36-\varepsilon}$ with $Q\leq X^{1/2+1/66-δ}$, along with wider bounds for $N$ if $Q\leq X^{\frac{45}{89}-ε}$. The former improves the allowable range of $N$, while the latter improves the range of $Q$. To prove these new bounds, we improve Bettin and Chandee's famous result on trilinear forms with Kloosterman fractions in the case where the denominator has a fixed factor.

math.NT↗
arXiv2025

A Shiu Theorem for Larger and Smoother Functions

In this paper, we broaden Shiu's Brun-Titchmarsh theorem to allow for functions that are larger and/or smooth-supported. In particular, let $f$ be a nonnegative multiplicative function. We prove that if there exists a $β<1$ such that $f(p^l)\ll (\log\log x)^{lβ}$ for every prime $p$ and every $l>1$, and if $f(n)\ll \max\{n^ε,(\log x)^ε\}$ for every $ε>0$, then $$\sum_{\substack{x\leq n\leq x+y \\ n\equiv a\pmod k}}f(n)\ll \frac{y}{ϕ(k)(\log x)^{1-ε_0}}\exp\left(\sum_{\substack{p\leq x \\ p\nmid k}}\frac{f(p)}{p}\right)$$ for every $ε_0>0$, where $x$, $y$, and $k$ are as they were in Shiu's original paper and $(a,k)=1$. Moreover, we prove that if $f$ is a $Q$-smooth-supported function then there exists a constant $C$ for which $$\sum_{\substack{x\leq n\leq x+y \\ n\equiv a\pmod k}}f(n)\ll \frac{y}{ϕ(k)(\log x)^{1-ε_0}}\exp\left(\sum_{\substack{p\leq x \\ p\nmid k}}\frac{f(p)}{p}\right)ρ(u)^C,$$ where $u=\frac{\log x}{\log Q}$, $ρ$ is the Dickman-de Bruijn function, and $C$ depends on whether we choose the bound of $f(p^l)\leq A_1^l$ or $f(p^l)\ll (\log\log x)^{lβ}$. We also give applications to both the divisor function to large powers and to smooth numbers in short intervals.

math.NT↗
arXiv2025

Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes

Let $χ$ be a Dirichlet character mod $D$ with $L(s,χ)$ its associated $L$-function, and let $ψ(x,q,a)$ be Chebyshev's prime-counting function for primes congruent to $a$ modulo $q$. We show that under the assumption of an exceptional character $χ$ with $L(1,χ)=o\left((\log D)^{-5}\right)$, for any $q<x^{\frac 23-\varepsilon}$, the asymptotic $$ψ(x,q,a)=\frac{ψ(x)}{ϕ(q)}\left(1-χ\left(\frac{aD}{(D,q)}\right)+o(1)\right)$$ holds for almost all $a$ with $(a,q)=1$. We also find that for any fixed $a$, the above holds for almost all $q<x^{\frac 23-\varepsilon}$ with $(a,q)=1$. Previous prime equidistribution results under the assumption of Siegel zeroes (by Friedlander-Iwaniec and the current author) have found that the above asymptotic holds either for all $a$ and $q$ or on average over a range of $q$ (i.e. for the Elliott-Halberstam conjecture), but only under the assumption that $q<x^θ$ where $θ=\frac{30}{59}$ or $\frac{16}{31}$, respectively.

math.NT↗
arXiv2024

Prime Distribution and Siegel Zeroes

Let $χ$ be a Dirichlet character mod $D$ with $L(s,χ)$ its associated $L$-function, and let $ψ(x,q,a)$ be, as usual, Chebyshev's prime-counting function for the primes of the arithmetic progression $a$ (mod $q$) with $(a,q)=1$. For a fixed $R>7$, we prove that under the assumption of an exceptional character $χ$ with $L(1,χ)<(\log D)^{-R}$, there exists a range of $x$ for which the asymptotic $$ψ(x,q,a)=\frac{ψ(x)}{ϕ(q)}\left(1-χ\left(\frac{aD}{(q,D)}\right)+o(1)\right)$$ holds for $q<x^{\frac{30}{59}-\varepsilon}$. We also show slightly better bounds for $q$ if we take an average over a range of $q$, finding an Elliott-Halberstam-type result for $q\sim Q$ on the range $Q<x^{\frac{16}{31}-\varepsilon}$. This improves on a Friedlander and Iwaniec 2003 result that requires $q<x^{\frac{233}{462}}$ and $R\geq 554,401^{554,401}$.

math.NT↗
arXiv2024

Carmichael numbers and least common multiples of $p-1$

For a Carmichael number $n$ with prime factors $p_1,\cdots,p_m$, define $$K=GCD[p_1-1,\cdots,p_m-1],$$ and let $C_ν(X)$ denote the number of Carmichael numbers up to $X$ such that $K=ν$. Assuming a strong conjecture on the first prime in an arithmetic progression, we prove that for any even natural number $ν$, $$C_ν(X)\geq X^{1-(2+o(1))\frac{\log\log\log \log X}{\log\log\log X}}.$$ This is a departure from standard constructions of Carmichael numbers, which generally require $K$ to grow along with $n$.

math.NT↗
arXiv2024

Carmichael Numbers with Prime Numbers of Prime Factors

Under the assumption of Heath-Brown's conjecture on the first prime in an arithmetic progression, we prove that there are infinitely many Carmichael numbers $n$ such that the number of prime factors of $n$ is prime.

math.NT↗
arXiv2023

Mitigating Modal Noise in Multimode Circular Fibres by Optical Agitation using a Galvanometer

Modal noise appears due to the non-uniform and unstable distribution of light intensity among the finite number of modes in multimode fibres. It is an important limiting factor in measuring radial velocity precisely by fibre-fed high-resolution spectrographs. The problem can become particularly severe as the fibre's core become smaller and the number of modes that can propagate reduces. Thus, mitigating modal noise in relatively small core fibres still remains a challenge. We present here a novel technique to suppress modal noise. Two movable mirrors in the form of a galvanometer reimage the mode-pattern of an input fibre to an output fibre. The mixing of modes coupled to the output fibre can be controlled by the movement of mirrors applying two sinusoidal signals through a voltage generator. We test the technique for four multimode circular fibres: 10 and 50 micron step-index, 50 micron graded-index, and a combination of 50 micron graded-index and 5:1 tapered fibres (GI50t). We present the results of mode suppression both in terms of the direct image of the output fibre and spectrum of white light obtained with the high-resolution spectrograph. We found that the galvanometer mitigated modal noise in all the tested fibres, but was most useful for smaller core fibres. However, there is a trade-off between the modal noise reduction and light-loss. The GI50t provides the best result with about 60% mitigation of modal noise at a cost of about 5% output light-loss. Our solution is easy to use and can be implemented in fibre-fed spectrographs.

astro-ph.IM↗
arXiv2023

Prime Tuples and Siegel Zeros

Under the assumption of infinitely many Siegel zeroes $s$ with $Re(s)>1-\frac{1}{(\log q)^{R}}$ for a sufficiently large value of $R$, we prove that there exist infinitely many $m$-tuples of primes that are $\ll e^{1.9828m}$ apart. This "improves" (in some sense) on the bounds of Maynard-Tao, Baker-Irving, and Polymath 8b, who found bounds of $e^{3.815m}$ unconditionally and $me^{2m}$ assuming the Elliott-Halberstam conjecture; it also generalizes a 1983 result of Heath-Brown that states that infinitely many Siegel zeroes would imply infinitely many twin primes. Under this assumption of Siegel zeroes, we also improve the upper bounds for the gaps between prime triples, quadruples, quintuples, and sextuples beyond the bounds found via Elliott-Halberstam.

math.NT↗
arXiv2020

Technical Report: Property-Directed Verified Monitoring of Signal Temporal Logic

Signal Temporal Logic monitoring over numerical simulation traces has emerged as an effective approach to approximate verification of continuous and hybrid systems. In this report we explore an exact verification procedure for STL properties based on monitoring verified traces in the form of Taylor model flowpipes as produced by the Flow* verified integrator. We explore how tight integration with Flow*'s symbolic flowpipe representation can lead to more precise and more efficient monitoring. We then show how the performance of monitoring can be increased substantially by introducing masks, a property-directed refinement of our method which restricts flowpipe monitoring to the time regions relevant to the overall truth of a complex proposition. Finally, we apply our implementation of these methods to verifying properties of a challenging continuous system, evaluating the impact of each aspect of our procedure on monitoring performance.

cs.LO↗
arXiv2018

The Bond-Calculus: A Process Algebra for Complex Biological Interaction Dynamics

We present the bond-calculus, a process algebra for modelling biological and chemical systems featuring nonlinear dynamics, multiway interactions, and dynamic bonding of agents. Mathematical models based on differential equations have been instrumental in modelling and understanding the dynamics of biological systems. Quantitative process algebras aim to build higher level descriptions of biological systems, capturing the agents and interactions underlying their behaviour, and can be compiled down to a range of lower level mathematical models. The bond-calculus builds upon the work of Kwiatkowski, Banks, and Stark's continuous pi-calculus by adding a flexible multiway communication operation based on affinity patterns and general kinetic laws. We develop a compositional semantics based on vector fields and linear operators, which we use to define the time evolution of this system. This enables simulation and analysis via differential equation generation or stochastic simulation. Finally, we apply our framework to an existing biological model: Kuznetsov's classic model of tumour immune interactions.

cs.LO↗
arXiv2016

There are infinitely many elliptic Carmichael numbers

In 1987, Dan Gordon defined an elliptic curve analogue to Carmichael numbers known as elliptic Carmichael numbers. In this paper, we prove that there are infinitely many elliptic Carmichael numbers. In doing so, we resolve in the affirmative the question of whether there exist infinitely square-free, composite integers $n$ such that for every prime $p$ that divides $n$, $p+1|n+1$.

math.NT↗
arXiv2015

Infinitude of $k$-Lehmer numbers which are not Carmichael

In this paper, we prove that there are infinitely many $n$ for which $rad(φ(n))|n-1$ but $n$ is not a Carmichael number. Additionally, we prove that for any $k\geq 3$, there exist infinitely many $n$ such that $φ(n)|(n-1)^k$ but $φ(n)\nmid (n-1)^{k-1}$. The constructs that we consider here are generalizations of Carmichael and Lehmer numbers, respectively, that were first formulated by Grau and Oller-Marcén.

math.NT↗
arXiv2014

Variants of Korselt's Criterion

Under sufficiently strong assumptions about the first term in an arithmetic progression, we prove that for any integer $a$, there are infinitely many $n\in \mathbb N$ such that for each prime factor $p|n$, we have $p-a|n-a$. This can be seen as a generalization of Carmichael numbers, which are integers $n$ such that $p-1|n-1$ for every $p|n$.

math.NT↗