Related papers: On the general additive divisor problem

Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as…

Finding a good bound on the maximal edge diameter $\Delta(d,n)$ of a polytope in terms of its dimension $d$ and the number of its facets $n$ is one of the basic open questions in polytope theory \cite{BG}. Although some bounds are known,…

Let $a(n)$ be the number of partitions of $n$ of the form $a_1 + a_2 + \cdots + a_k$ where $a_{i + 1}$ is a proper divisor of $a_i$ for all $i < k$. Erd{\H o}s and Loxton showed that the sum of $a(n)$ over all $n \leq x$ is asymptotic to a…

Estimating the discrepancy of the hypergraph of all arithmetic progressions in the set $[N]=\{1,2,\hdots,N\}$ was one of the famous open problems in combinatorial discrepancy theory for a long time. An extension of this classical hypergraph…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)$ with $\Delta^*(x) = -\Delta(x) +…

For integer $n\geqslant 1$ and real $u$, let $\Delta(n,u):=|\{d:d\mid n,\,{\rm e}^u<d\leqslant {\rm e}^{u+1}\}|$. The Erd\H{o}s--Hooley Delta-function is then defined by $\Delta(n):=\max_{u\in{\mathbb R}}\Delta(n,u).$ We improve the current…

Let $\Delta(x)$ be the error term of the Dirichlet divisor problem. An asymptotic formula with the error term $O(T^{53/28+\epsilon})$ is established for the integral $\int_1^T\Delta^4(x)dx.$ Similar results are also established for some…

We prove an upper bound for the exponential sum associated to a localized $k-$divisor function, i.e., the counting function of the number of ways to write a positive integer $n$ as a product of $k\ge 2$ positive integers, each of them…

When $s\ge k\ge 3$ and $n_1,\ldots ,n_k$ are large natural numbers, denote by $A_{s,k}(\mathbf n)$ the number of solutions in non-negative integers $\mathbf x$ to the system \[ x_1^j+\ldots +x_s^j=n_j\quad (1\le j\le k). \] Under…

Nguyen has shown that on averaging over $a=1,...,q$ the 3-fold divisor function has exponent of distribution 2/3, following \cite {banks}. We follow [2] which leads to stronger bounds.

Let $A$ be an additive basis of order $h$ and $X$ be a finite nonempty subset of $A$ such that the set $A \setminus X$ is still a basis. In this article, we give several upper bounds for the order of $A \setminus X$ in function of the order…

Given a multiplicative function $f$, we let $S(x,f)=\sum_{n\leq x}f(n)$ be the associated partial sum. In this note, we show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums…

Let $d^{(k)}(n)$ be the $k$-free divisor function for integer $k\ge2$. Let $a$ be a nonzero integer. In this paper, we establish an asymptotic formula \begin{equation*} \sum_{p\leq x} d^{(k)}(p-a) =b_k \cdot x+O\left(\frac{x}{\log x}\right)…

In this paper, we investigate a weighted divisor problem involving the exponential sum of $D_{(1)}(n)$, the $n$th coefficient in the Dirichlet series expansion of $\zeta'(s)^2$. We establish a truncated Vorono\"{i} type formula for the…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x)…

Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq…

Let n(2,k) denote the largest integer n for which there exists a set A of k nonnegative integers such that the sumset 2A contains {0,1,2,...,n-1}. A classical problem in additive number theory is to find an upper bound for n(2,k). In this…

We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture…

We study averages along the integers using the divisor function $d(n)$, and defined as $$K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) \,f(x+n) , $$ where $D(N) = \sum _{n=1} ^N d(n) $. We shall show that these averages satisfy a…

For a fixed $\theta\neq 0$, we define the twisted divisor function $$ \tau(n, \theta):=\sum_{d\mid n}d^{i\theta}\ .$$ In this article we consider the error term $\Delta(x)$ in the following asymptotic formula $$ \sum_{n\leq x}^*|\tau(n,…