Related papers: On a Conjecture by Ren and Li

In 1971 Cusick proved that every real number $x\in[0,1]$ can be expressed as a sum of two continued fractions with no partial quotients equal to $1$. In other words, if we define a set $$ S(k):= \{ x\in[0,1] : a_n(x) \geq k \text{ for all }…

We state and prove an equivariant version of Lehmer's conjecture on heights, generalizing papers by Zagier (1993) and Dresden (1998) which are special cases of this theorem. We also extend their three cases to a full classification of all…

Consider the equation div$(\varphi^2 \nabla \sigma)=0$ in $\mathbb{R}^N,$ where $\varphi>0$. Berestycki, Caffarelli and Nirenberg proved that if there exists $C>0$ such that $\int_{B_R}(\varphi \sigma)^2 \leq CR^2$ for every $R\geq 1$ then…

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x(log x)^{-1+o(1)} numbers not exceeding x common to the ranges of Euler's function phi(n) and the sum-of-divisors function sigma(m).

A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is…

We provide new sufficient conditions under which Ryser's conjecture holds.

In a recent paper, Svante Janson has considered a conjecture suggested by Va\v{s}ek Chv\a'atal dealing with the probability that a binomial random variable with parameters $n$ and $m/n$ - where $m$ is an integer - exceeds its expectation…

Given $r \in \mathbb{N}$, define the function $S_{r}: \mathbb{N} \rightarrow \mathbb{Q}$ by $S_{r}(n)=\displaystyle \sum_{k=0}^{n} \frac{k}{k+r} \binom{n}{k}$. In $2015$, the second author conjectured that there are infinitely many $r \in…

Let $M$ be a compact 3-dimensional Riemannian manifold with nonnegative Ricci curvature and a nonempty boundary $\partial M$. Fraser and Li \cite{Fraser&Li} established a compactness theorem for the space of compact, properly embedded…

In \cite{reed97}, Reed conjectures that the inequality $\chi (G) \leq \left \lceil \textstyle {1/2} (\omega (G) + \Delta (G) + 1) \right \rceil$ holds for any graph $G$. We prove this holds for a graph $G$ if $\bar{G}$ is disconnected. From…

This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $d\mu = w(\theta) \frac{d\theta}{2\pi} + d\mu_s$ with Verblunsky coefficients…

Suppose that we have a set of numbers x_1, ..., x_n which have nonnegative sum. How many subsets of k numbers from {x_1, ..., x_n} must have nonnegative sum? Manickam, Miklos, and Singhi conjectured that for n at least 4k the answer is (n-1…

Let $(R,m)\to (S,n)$ be a flat local extension of local rings. Lech conjectured in 1960 that there should be a general inequality $e(R)\leq e(S)$ on the Hilbert-Samuel multiplicities. This conjecture is known when the base ring $R$ has…

Let $X_1$, $X_2$, $\ldots$, $X_n$ be a sequence of coherent random variables, i.e., satisfying the equalities $$ X_j=\mathbb{P}(A|\mathcal{G}_j),\qquad j=1,\,2,\,\ldots,\,n,$$ almost surely for some event $A$. The paper contains the proof…

In this paper, we make use of Robin and Lagarias' criteria to prove Riemann hypothesis. The goal is, using Lagarias criterion for $n\geq 1$ since Lagarias criterion states that Riemann hypothesis holds if and only if the inequality…

We give a new proof of the Smale conjecture for $\mathbb{RP}^3$ and all lens spaces using minimal surfaces and min-max theory. For $\mathbb{RP}^3$, the conjecture was first proved in 2019 by Bamler-Kleiner using Ricci flow.

Peng (2006) initiated a new kind of central limit theorem under sub-linear expectations. Song (2017) gave an estimate of the rate of convergence of Peng's central limit theorem. Based on these results, we establish a new kind of almost sure…

The results of the study provide guidelines for the development and applications of algorithms. When the number of steps for calculating an assumption tends to infinity, probability theory can be applied to predict whether the assumption…

Let $k\geq 2$ be an integer and let $\lambda$ be the Liouville function. Given $k$ non-negative distinct integers $h_1,\ldots,h_k$, the Chowla conjecture claims that $\sum_{n\leq x}\lambda(n+h_1)\cdots \lambda(n+h_k)=o(x)$ as $x\to\infty$.…

We prove that the Riemann hypothesis is equivalent to the condition $\int_{2}^x\left(\pi(t)-\text{li}(t)\right)\mathrm{d}t<0$ for all $x>2$. Here, $\pi(t)$ is the prime-counting function and $\text{li}(t)$ is the logarithmic integral. This…