Related papers: On the Least counterexample to Robin hypothesis

For a graph $G$, its energy $\mathcal{E}(G)$ is the sum of absolute values of the eigenvalues of its adjacency matrix, the matching number $\mu(G)$ is the number of edges in a maximum matching of $G$, while $\Delta$ is the maximum vertex…

Inspired by Cohen and te Riele~\cite{Cohen1996}, who computationally verified that for every $n \leq 400$ there exists $k$ such that $\sigma^k(n) \equiv 0 \pmod{n}$ (where $\sigma^k$ denotes the $k$-fold iteration of the sum-of-divisors…

Consider the equation div$(\varphi^2 \nabla \sigma)=0$ in $\mathbb{R}^N,$ where $\varphi>0$. It is well-known that if there exists $C>0$ such that $\int_{B_R}(\varphi \sigma)^2 dx\leq CR^2$ for every $R\geq 1$ then $\sigma$ is necessarily…

Ramanujan proved that the inequality $\pi(x)^2 < \frac{e x}{\log x} \pi\Big(\frac{x}{e}\Big)$ holds for all sufficiently large values of $x$. Using an explicit estimate for the error in the prime number theorem, we show unconditionally that…

If $G$ is a finite Abelian group, define $s_{k}(G)$ to be the minimal $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. Recently Bitz et al. proved that if $n = exp(G)$, then…

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph…

We show that for any natural number $s$, there is a constant $\gamma$ and a subgraph-closed class having, for any natural $n$, at most $\gamma^n$ graphs on $n$ vertices up to isomorphism, but no adjacency labeling scheme with labels of size…

This article defines a new type of abundant numbers, called largest rho-value (abbreviate LR) numbers, and then shows that Robin hypothesis is true if and only if all LR numbers $>5040$ satisfy Robin inequality.

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this note, we prove that if $\sigma_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have $\sigma_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$.…

For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$ that have equal multisets of isomorphism types of proper subgroups and proper quotient groups, isomorphic character tables, and power maps. This obstructs recent…

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$…

We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $\gcd(n,\phi(n))=1$. With $C(x)$ denoting the count of cyclic $n\le x$,…

Consider the boundary case in a one-dimensional super-critical branching random walk. It is known that upon the survival of the system, the minimal position after $n$ steps behaves in probability like ${3\over 2} \log n$ when $n\to \infty$.…

The Ramsey number $R(s,t)$ is the smallest integer $n$ such that all graphs of size $n$ contain a clique of size $s$ or an independent set of size $t$. $\mathcal{R}(s,t,n)$ is the set of all counterexample graphs without this property for a…

Let $f(N)$ denote the least integer $k$ such that, if $G$ is an abelian group of order $N$ and $A \subseteq G$ is a uniformly random $k$-element subset, then with probability at least $\tfrac12$ the subset-sum set $\{ \sum_{x \in S} x : S…

In a recent work [JNT \textbf{129}, 2154 (2009)], Gun and co-workers have claimed that the number $\,\log{\Gamma(x)} + \log{\Gamma(1-x)}\,$, $x$ being a rational number between $0$ and $1$, is transcendental with at most \emph{one} possible…

Let $\vfi$ be Euler's function, $\ga$ be Euler's constant and $N_k$ be the product of the first $k$ primes. In this article, we consider the function $c(n) =(n/\vfi(n)-e^\ga\log\log n)\sqrt{\log n}$. Under Riemann's hypothesis, it is proved…

We prove a tight quantum query lower bound $\Omega(n^{k/(k+1)})$ for the problem of deciding whether there exist $k$ numbers among $n$ that sum up to a prescribed number, provided that the alphabet size is sufficiently large. This is an…

"The Baron's omni-sequence", B(n), first defined by Khovanova and Lewis (2011), is a sequence that gives for each n the minimum number of weighings on balance scales that can verify the correct labeling of n identically-looking coins with…

Let $P$ be a finite set of points in the plane in general position, that is, no three points of $P$ are on a common line. We say that a set $H$ of five points from $P$ is a $5$-hole in $P$ if $H$ is the vertex set of a convex $5$-gon…