Solutions to some sign change problems on the functions involving sums of divisors
In this note, we solve some sign change problems on the functions involving sums of divisors posed by Pongsriiam recently.
math.NT↗
arXiv subjects
Yuchen Ding
Publications and source records attributed to Yuchen Ding.
On a conjecture of R. M. Murty and V. K. Murty II
Let $ω^*(n)$ be the number of primes $p$ such that $p-1$ divides $n$. Assuming the Elliott--Halberstam Conjecture, we prove a conjecture posted by M. R. Murty and V. K. Murty in 2021 which states that $$\sum_{n\leqslant x}ω^*(n)^2\sim 2\frac{ζ(2)ζ(3)}{ζ(6)}x\log x, \quad \text{as} \quad x\rightarrow \infty.$$ The above sum was first investigated by Prachar in 1955. One of the key ingredients in our argument is the application of a sieve result on estimating various certain summations involving primes in arithmetic progressions, rather than a direct use of the Brun--Titchmarsh inequality which would not be applicable for our task.
math.NT↗
On a conjecture of Ram\'ırez Alfons\'ın and Skałba III
Let $1<c<d$ be two relatively prime integers. For a non-negative integer $\ell$, let $g_\ell(c,d)$ be the largest integer $n$ such that $n=c x+d y$ has at most $\ell$ non-negative solutions $(x,y)$. In this paper we prove that $$ π_{\ell,c,d}\sim\frac{π\bigl(g_\ell(c,d)\bigr)}{2 \ell+2}\quad(\text{as}~ c\to\infty)\,, $$ where $π_{\ell,c,d}$ is the number of primes $n$ having more than $\ell$ distinct non-negative solutions to $n=c x+d y$ with $n\le g_\ell(c,d)$, and $π(x)$ denotes the number of all primes up to $x$. The case where $\ell=0$ has been proved by Ding, Zhai and Zhao recently, which was conjectured formerly by Ram\'ırez Alfons\'ın and Skałba.
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On a conjecture of Ram\'ırez Alfons\'ın and Skałba II
Let $1<c<d$ be two relatively prime integers and $g_{c,d}=cd-c-d$. We confirm, by employing the Hardy--Littlewood method, a 2020 conjecture of Ram\'ırez Alfons\'ın and Skałba which states that $$#\left\{p\le g_{c,d}:p\in \mathcal{P}, ~p=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0}\right\}\sim \frac{1}{2}π\left(g_{c,d}\right) \quad (\text{as}~c\rightarrow\infty),$$ where $\mathcal{P}$ is the set of primes, $\mathbb{Z}_{\geqslant0}$ is the set of nonnegative integers and $π(t)$ denotes the number of primes not exceeding $t$.
math.NT↗
Correct order on some certain weighted representation functions
Let $\mathbb{N}$ be the set of all nonnegative integers. For any positive integer $k$ and any subset $A$ of nonnegative integers, let $r_{1,k}(A,n)$ be the number of solutions $(a_1,a_2)$ to the equation $n=a_1+ka_2$. In 2016, Qu proved that $$\liminf_{n\rightarrow\infty}r_{1,k}(A,n)=\infty$$ providing that $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ for all sufficiently large integers, which answered affirmatively a 2012 problem of Yang and Chen. In a very recent article, another Chen (the first named author) slightly improved Qu's result and obtained that $$\liminf_{n\rightarrow\infty}\frac{r_{1,k}(A,n)}{\log n}>0.$$ In this note, we further improve the lower bound on $r_{1,k}(A,n)$ by showing that $$\liminf_{n\rightarrow\infty}\frac{r_{1,k}(A,n)}{n}>0.$$ Our bound reflects the correct order of magnitude of the representation function $r_{1,k}(A,n)$ under the above restrictions due to the trivial fact that $r_{1,k}(A,n)\le n/k.$
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A new upper bound on Ruzsa's number on the Erd\H os-Turán conjecture
In this note, we show that the Ruzsa number $R_m$ is bounded by $192$ for any positive integer $m$, which improved the prior bound $R_m\le 288$ given by Y.--G. Chen in 2008.
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Solution to a problem of Luca, Menares and Pizarro-Madariaga
Let $k\ge 2$ be a positive integer and $P^+(n)$ the greatest prime factor of a positive integer $n$ with convention $P^+(1)=1$. For any $θ\in \left[\frac 1{2k},\frac{17}{32k}\right)$, set $$T_{k,θ}(x)=\sum_{\substack{p_1\cdot\cdot\cdot p_k\le x\\ P^+(\gcd(p_1-1,...,p_k-1))\ge (p_1\cdot\cdot\cdot p_k)^θ}}1,$$ where the $p'$s are primes. It is proved that $$T_{k,θ}(x)\ll_{k}\frac{x^{1-θ(k-1)}}{(\log x)^2},$$ which, together with the lower bound $$T_{k,θ}(x)\gg_{k}\frac{x^{1-θ(k-1)}}{(\log x)^2}$$ obtained by Wu in 2019, answer a 2015 problem of Luca, Menares and Pizarro-Madariaga on the exact order of magnitude of $T_{k,θ}(x)$. A main novelty in the proof is that, instead of using the Brun--Titchmarsh theorem to estimate the $k^{th}$ movement of primes in arithmetic progressions, we transform the movement to an estimation involving taking primes simultaneously by linear shifts of primes.
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On three problems of Y.--G. Chen
In this short note, we answer two questions of Chen and Ruzsa negatively and answer a problem of Ma and Chen affirmatively.
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A Thorough Examination on Zero-shot Dense Retrieval
Recent years have witnessed the significant advance in dense retrieval (DR) based on powerful pre-trained language models (PLM). DR models have achieved excellent performance in several benchmark datasets, while they are shown to be not as competitive as traditional sparse retrieval models (e.g., BM25) in a zero-shot retrieval setting. However, in the related literature, there still lacks a detailed and comprehensive study on zero-shot retrieval. In this paper, we present the first thorough examination of the zero-shot capability of DR models. We aim to identify the key factors and analyze how they affect zero-shot retrieval performance. In particular, we discuss the effect of several key factors related to source training set, analyze the potential bias from the target dataset, and review and compare existing zero-shot DR models. Our findings provide important evidence to better understand and develop zero-shot DR models.
cs.CL↗
Quantitative results of the Romanov type representation functions
For $α>0$, let $$\mathscr{A}=\{ a_1 (\log m)^α$ for infinitely many positive integers $m$ and $\ell_m<0.9\log\log m$ for sufficiently integers $m$. Suppose further that $(\ell_i,a_i)=1$ for all $i$. For any $n$, let $f_{\mathscr{A},\mathscr{L}}(n)$ be the number of the available representations listed below $$\ell_in=p+a_i \quad \left(1\le i\le \mathscr{A}(n)\right),$$ where $p$ is a prime number. It is proved that $$\limsup_{n\to \infty } \frac{f_{\mathscr{A},\mathscr{L}}(n)}{\log\log n}>0,$$ which covers an old result of Erd\H os in 1950 by taking $a_i=2^i$ and $\ell_i=1$. One key ingredient in the argument is a technical lemma established here which illustrates how to pick out the admissible parts of an arbitrarily given set of distinct linear functions. The proof then reduces to the verifications of a hypothesis involving well--distributed sets introduced by Maynard, which of course would be the other key ingredient in the argument.
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A note on additive complements of the squares
Let $\mathcal{S}=\{1^2,2^2,3^2,...\}$ be the set of squares and $\mathcal{W}=\{w_n\}_{n=1}^{\infty} \subset \mathbb{N}$ be an additive complement of $\mathcal{S}$ so that $\mathcal{S} + \mathcal{W} \supset \{n \in \mathbb{N}: n \geq N_0\}$ for some $N_0$. Let $\mathcal{R}_{\mathcal{S},\mathcal{W}}(n) = \#\{(s,w):n=s+w, s\in \mathcal{S}, w\in \mathcal{W}\} $. In 2017, Chen-Fang \cite{C-F} studied the lower bound of $\sum_{n=1}^NR_{\mathcal{S},\mathcal{W}}(n)$. In this note, we improve Cheng-Fang's result and get that $$\sum_{n=1}^NR_{\mathcal{S},\mathcal{W}}(n)-N\gg N^{1/2}.$$ As an application, we make some progress on a problem of Ben Green problem by showing that $$\limsup_{n\rightarrow\infty}\frac{\frac{π^2}{16}n^2-w_n}{n}\ge \fracπ{4}+\frac{0.193π^2}{8}.$$
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Vetaverse: A Survey on the Intersection of Metaverse, Vehicles, and Transportation Systems
Since 2021, the term "Metaverse" has been the most popular one, garnering a lot of interest. Because of its contained environment and built-in computing and networking capabilities, a modern car makes an intriguing location to host its own little metaverse. Additionally, the travellers don't have much to do to pass the time while traveling, making them ideal customers for immersive services. Vetaverse (Vehicular-Metaverse), which we define as the future continuum between vehicular industries and Metaverse, is envisioned as a blended immersive realm that scales up to cities and countries, as digital twins of the intelligent Transportation Systems, referred to as "TS-Metaverse", as well as customized XR services inside each Individual Vehicle, referred to as "IV-Metaverse". The two subcategories serve fundamentally different purposes, namely long-term interconnection, maintenance, monitoring, and management on scale for large transportation systems (TS), and personalized, private, and immersive infotainment services (IV). By outlining the framework of Vetaverse and examining important enabler technologies, we reveal this impending trend. Additionally, we examine unresolved issues and potential routes for future study while highlighting some intriguing Vetaverse services.
cs.HC↗
On a conjecture on shifted primes with large prime factors
Let $\mathcal{P}$ be the set of all primes and $π(x)$ be the number of primes up to $x$. For any $n\ge 2$, let $P^+(n)$ be the largest prime factor of $n$. For $0<c<1$, let $$T_c(x)=\#\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\}.$$ In this note, we proved that there exists some $c<1$ such that $$\limsup_{x\rightarrow\infty}\frac{T_c(x)}{π(x)}<\frac12,$$ which disproves a conjecture of Chen and Chen.
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On a conjecture of R. M. Murty and V. K. Murty
Let $ω^*(n)$ be the number of primes $p$ such that $p-1$ divides $n$. Recently, R. M. Murty and V. K. Murty proved that $$x(\log\log x)^3\ll\sum_{n\le x}ω^*(n)^2\ll x\log x.$$ They further conjectured that there is some positive constant $C$ such that $$\sum_{n\le x}ω^*(n)^2\sim Cx\log x$$ as $x\rightarrow \infty$. In this short note, we give the correct order of the sum by showing that $$\sum_{n\le x}ω^*(n)^2\asymp x\log x.$$
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A counterexample of two Romanov type conjectures
In this note, we disprove two Romanov type conjectures posed by Chen.
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On the robustness of triaxial Schwarzschild modelling: The effects of correcting the orbit mirroring
In the past 15 years, the triaxial Schwarzschild orbit-superposition code by van den Bosch et al. (2008) has been widely applied to study the dynamics of galaxies. Recently, Quenneville et al. (2022) reported a bug in the orbit calculation of this code, specifically in the mirroring procedure that is used to speed up the computation. We have fixed the incorrect mirroring in DYNAMITE, which is the publicly-released successor of the triaxial Schwarzschild code by van den Bosch et al. (2008). In this study, we provide a thorough quantification of how this bug has affected the results of dynamical analyses performed with this code. We compare results obtained with the original and corrected versions of DYNAMITE, and discuss the differences in the phase-space distribution of a single orbit and in the global stellar orbit distribution, in the mass estimate of the central black hole in the highly triaxial galaxy PGC 46832, and in the measurement of intrinsic shape and enclosed mass for more than 50 galaxies. Focusing on the typical scientific applications of a Schwarzschild triaxial code, in all our tests we find that differences are negligible with respect to the statistical and systematic uncertainties. We conclude that previous results with the van den Bosch et al. (2008) triaxial Schwarzschild code are not significantly affected by the incorrect mirroring.
astro-ph.GA↗
Sum of elements in finite Sidon sets II
A set $S\subset\{1,2,...,n\}$ is called a Sidon set if all the sums $a+b~~(a,b\in S)$ are different. Let $S_n$ be the largest cardinality of the Sidon sets in $\{1,2,...,n\}$. In a former article, the author proved the following asymptotic formula $$\sum_{a\in S,~|S|=S_n}a=\frac{1}{2}n^{3/2}+O(n^{111/80+\varepsilon}),$$ where $\varepsilon>0$ is an arbitrary small constant. In this note, we give an extension of the above formula. We show that $$\sum_{a\in S,~|S|=S_n}a^{\ell}=\frac{1}{\ell+1}n^{\ell+1/2}+O\left(n^{\ell+61/160}\right)$$ for any positive integers $\ell$. Besides, we also consider the asymptotic formulae of other type summations involving Sidon sets. The proofs are established in a more general setting, namely we obtain the asymptotic formulae of the Sidon sets with $t$ elements when $t$ is near the magnitude $n^{1/2}$.
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On a problem of Romanoff type
Let $\mathcal{P}$ and $\mathbb{N}$ be the sets of all primes and natural numbers, respectively. In this article, it is proved that there has a positive lower density of the natural numbers which can be represented by the form $$p+2^{m_1^2}+2^{m_2^2},p\in \mathcal{P},m_1,m_2\in \mathbb{N}.$$ This solves a problem of Chen and Yang in 2014.
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