Related papers: Correct order on some certain weighted representat…

An $(n,k)$ sequence covering array is a set of permutations of $[n]$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of at least one of the permutations. An $(n,k)$ sequence covering array is perfect if there is a…

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let the representation function $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s<s'$. In…

For a set of positive integers $A$, let $p_A(n)$ denote the number of ways to write $n$ as a sum of integers from $A$, and let $p(n)$ denote the usual partition function. In the early 40s, Erd\H{o}s extended the classical Hardy--Ramanujan…

Let $K$ be a totally real number field of degree $n$ over $\mathbb{Q}$, with discriminant and regulator $\Delta_K, R_K$ respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary…

We consider some coloring issues related to the famous Erd\H {o}s Discrepancy Problem. A set of the form $A_{s,k}=\{s,2s,\dots,ks\}$, with $s,k\in \mathbb{N}$, is called a \emph{homogeneous arithmetic progression}. We prove that for every…

We present an extension of the Hardy--Littlewood inequality for multilinear forms. More precisely, let $\mathbb{K}$ be the real or complex scalar field and $m,k$ be positive integers with $m\geq k\,$ and $n_{1},\dots ,n_{k}$ be positive…

Under appropriate positivity hypotheses, we prove quantitative estimates for the total $k$-th order $Q$-curvature functional near minimizing metrics on any smooth, closed $n$-dimensional Riemannian manifold for every integer $1 \leq k <…

For a set of nonnegative integers $S$ let $R_{S}(n)$ denote the number of unordered representations of the integer $n$ as the sum of two different terms from $S$. In this paper we focus on partitions of the natural numbers into two sets…

Refining an estimate of Croot, Dobbs, Friedlander, Hetzel and Pappalardi, we show that for all $k \geq 2$, the number of integers $1 \leq a \leq n$ such that the equation $a/n = 1/m_1 + \dotsc + 1/m_k$ has a solution in positive integers…

We obtain an upper bound for the sum $\sum_{n\leq N} (a_{n}/\varphi (a_{n}))^{s}$, where $\varphi$ is Euler's totient function, $s\in \mathbb{N}$, and $a_{1},\ldots, a_{N}$ are positive integers (not necessarily distinct) with some…

Following Wigert, various authors, including Ramanujan, Gronwall, Erd\H{o}s, Ivi\'{c}, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's result $$\max_{n\leq x} \log d(n) =…

In this short note, we provide a new infinite family of $K_{2, t+1}$-free graphs for each prime power $t$. Using these graphs, we show that it is possible to partition the edges of $K_n$ into parts, such that each part is isomorphic to our…

Given positive integers $a_1,...,a_n$ with $\gcd(a_1,...,a_n) = 1$, we call an integer t representable if there exist nonnegative integers $m_1,...,m_n$ such that $t = m_1 a_1 + ... + m_n a_n$. In this paper, we discuss the linear…

Let $k$ and $N$ be positive integers with $k\ge2$ even. In this paper we give general explicit upper-bounds in terms of $k$ and $N$ from which all the residual representations $\bar{\rho}_{f,\lambda}$ attached to non-CM newforms of weight…

It was asked by E. Szemer\'edi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each…

For any positive integer $n$, let $\sigma(n)=\sum_{d\mid n} d$. In 2020, M. Kobayashi and T. Trudgian showed that the natural density of positive integers n with $\sigma(kn+r_1) \geq \sigma(kn+r_2)$ is between 0.053 and 0.055. In this…

A $\{00,01,10,11\}$-valued function on the vertices of the $n$-cube is called a $t$-resilient $(n,2)$-function if it has the same number of $00$s, $01$s, $10$s and $11$s among the vertices of every subcube of dimension $t$. The Friedman and…

For a $k$-uniform hypergraph $F$ and positive integers $s$ and $N$, the generalized Erd\H{o}s-Rogers function $f^{(k)}_{F,s}(N)$ denotes the largest integer $m$ such that every $K_s^{(k)}$-free $k$-graph on $N$ vertices contains an $F$-free…

We prove that for all positive integers $n$ and $k$, there exists an integer $N = N(n,k)$ satisfying the following. If $U$ is a set of $k$ direction vectors in the plane and $\mathcal{J}_U$ is the set of all line segments in direction $u$…

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial-boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem…