On a conjecture of R. M. Murty and V. K. Murty II

Let $\omega^*(n)$ be the number of primes $p$ such that $p-1$ divides $n$. In 1955, Prachar proved that $\sum_{n\le x}\omega^*(n)^2=O\left(x(\log x)^2\right)$. Recently, Murty and Murty improved this to $$x(\log\log x)^3\ll\sum_{n\le x}\omega^*(n)^2\ll x\log x.$$ They further conjectured that there is some positive constant $C$ such that $$\sum_{n\le x}\omega^*(n)^2\sim Cx\log x$$ as $x\rightarrow \infty$. In a former note, the author gave the correct order of it by showing that $$\sum_{n\le x}\omega^*(n)^2\asymp x\log x.$$ In this subsequent article, we provide a conditional proof of their conjecture

A new upper bound on Ruzsa's number on the Erd\H os--Tur\'{a}n conjecture

In this note, we show that the Ruzsa number $R_m$ is bounded by $192$ for any positive integer $m$, which improved slightly the prior bound $R_m\le 288$ given by Y.--G. Chen in 2008.Comment: Comparing with former versions, some more related conjectures are posed at the end of the articl

Rumba : a Python framework for automating large-scale recursive internet experiments on GENI and FIRE+

It is not easy to design and run Convolutional Neural Networks (CNNs) due to: 1) finding the optimal number of filters (i.e., the width) at each layer is tricky, given an architecture; and 2) the computational intensity of CNNs impedes the deployment on computationally limited devices. Oracle Pruning is designed to remove the unimportant filters from a well-trained CNN, which estimates the filters’ importance by ablating them in turn and evaluating the model, thus delivers high accuracy but suffers from intolerable time complexity, and requires a given resulting width but cannot automatically find it. To address these problems, we propose Approximated Oracle Filter Pruning (AOFP), which keeps searching for the least important filters in a binary search manner, makes pruning attempts by masking out filters randomly, accumulates the resulting errors, and finetunes the model via a multi-path framework. As AOFP enables simultaneous pruning on multiple layers, we can prune an existing very deep CNN with acceptable time cost, negligible accuracy drop, and no heuristic knowledge, or re-design a model which exerts higher accuracy and faster inferenc

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 $\theta\in \left[\frac 1{2k},\frac{17}{32k}\right)$, set $$T_{k,\theta}(x)=\sum_{\substack{p_1\cdot\cdot\cdot p_k\le x\\ P^+(\gcd(p_1-1,...,p_k-1))\ge (p_1\cdot\cdot\cdot p_k)^\theta}}1,$$ where the $p'$s are primes. It is proved that $$T_{k,\theta}(x)\asymp_{k}\frac{x^{1-\theta(k-1)}}{(\log x)^2},$$ which answers a 2015 problem of Luca, Menares and Pizarro-Madariaga on the exact order of magnitude of $T_{k,\theta}(x)$. 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

On a conjecture of Ram\'{\i}rez Alfons\'{\i}n and Ska{\l}ba 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\'{\i}rez Alfons\'{\i}n and Ska{\l}ba 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}\pi\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 $\pi(t)$ denotes the number of primes not exceeding $t$