Related papers: Properties of counterexample to Robin hypothesis

We prove that for every $\epsilon > 0$ there exists a $\delta > 0$ so that every group of order $n \geq 3$ has at least $\delta \log_{2} n/{(\log_{2} \log_{2} n)}^{3+\epsilon}$ conjugacy classes. This sharpens earlier results of Pyber and…

Let $r(k,A,n)$ denote the number of representations of $n$ as a sum of $k$ elements of a set $A \subseteq \mathbb{N}$. In 2002, Dombi conjectured that if $A$ is co-infinite, then the sequence $(r(k,A,n))_{n \geq 0}$ cannot be strictly…

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such…

In 2017, Geordie Williamson proved the existence of counterexamples to James's conjecture on the decomposition matrices of symmetric groups and their Hecke algebras. The smallest counterexample detectable by Williamson's method occurs in…

Given a compact subset $\Sigma$ of the real numbers obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in $\Sigma$. The distribution of conjugates of such an integer defines a probability…

Let $C(n)$ denote the number of permutations $\sigma$ of $[n]=\{1,2,\dots,n\}$ such that $\gcd(j,\sigma(j))=1$ for each $j\in[n]$. We prove that for $n$ sufficiently large, $n!/3.73^n < C(n) < n!/2.5^n$.

It is shown that every sufficiently large even integer is a sum of two primes and exactly 13 powers of 2. Under the Generalized Rieman Hypothesis one can replace 13 by 7. Unlike previous work on this problem, the proof avoids numerical…

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb P(X_1=1)=\mathbb P(X_1=-1)=1/2$. Let $F(\sigma)=\sum_{n=1}^\infty X_nn^{-\sigma}$. We prove that the following holds almost surely…

Let $N$ be an odd perfect number. Let $\omega(N)$ be the number of distinct prime factors of $N$ and let $\Omega(N)$ be the total number of prime factors of $N$. We prove that if $(3,N)=1$, then $ \frac{302}{113}\omega - \frac{286}{113}…

For $k\mid n$ let $Comb_{n,k}$ denote the tree consisting of an $(n/k)$-vertex path with disjoint $k$-vertex paths beginning at each of its vertices. An old conjecture says that for any $k=k(n)$ the threshold for the random graph $G(n,p)$…

In this paper, we study the distribution of the sequence of integers $2^{\omega(n)}$ under the assumption of the strong Riemann hypothesis, where $\omega(n)$ denotes the number of distinct prime divisors of $n$. We provide an asymptotic…

Let $\phi(n)$ be the Euler totient function and $\sigma(n)$ denote the sum of divisors of $n$. In this note, we obtain explicit upper bounds on the number of positive integers $n\leq x$ such that $\phi(\sigma(n)) > cn$ for any $c>0$. This…

We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby…

Let $E$ be an elliptic curve over the finite field $\mathbb F_q$. We prove that, when $n$ is a sufficiently large positive integer, $\#E(\mathbb F_{q^n})$ has a prime factor exceeding $n\exp(c\log n/\log\log n)$.

This work is a probabilistic study of the 'primes' of the Cram\'er model. We prove that there exists a set of integers $\mathcal S$ of density 1 such that \begin{equation}\liminf_{ \mathcal S\ni n\to\infty} (\log n)\mathbb{P} \{S_n\…

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density…

We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists…

Let $n$ be a positive integer and let $S$ be a sequence of $n$ integers in the interval $[0,n-1]$. If there is an $r$ such that any nonempty subsequence with sum $\equiv 0$ $\pmod n$ has length $=r,$ then $S$ has at most two distinct…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this paper, we prove that if $\sigma_1(G)<2+\frac{11}{|G|}$\,, then $G$ is supersolvable. In particular, some new characterizations of the well-known groups…

An equivalence of the Riemann Hypothesis (RH) enables a direct bridge to the Young lattice. In specific, the classical threshold $\lim_{n\to\infty} \sigma(n)/(n \log\log n) = e^{\gamma} \approx 1.78107$, derived from the asymptotic behavior…