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We study the problem of obtaining asymptotic formulas for the sums $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$ and $\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$, where $\Lambda$ is the von Mangoldt function, $d_k$ is the $k^{\operatorname{th}}$…

Let $k \ge 2$ be a fixed integer. We define the multiplicative function $D_k(n) = d_k(n)/d_k^*(n)$, such that $d_k(n)$ is the Piltz divisor function and $d_k^*(n) = k^{\omega(n)}$ is its unitary analogue, where $\omega(n)$ is the number of…

Let $\mathcal{A}(n)$ be the $(1,n)-th$ Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke-Maass cusp form i.e. $\Lambda(1,n)$ or the triple divisor function $d_3(n)$, which is the number of solutions of the equation $r_1r_2r_3 = n$ with $r_1,…

Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. In this paper, we prove that $$\sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+\mathcal{O}_{\varepsilon,c}…

Consider the real numbers $$ \ell_{n,k} = \ln\left( \tfrac{3}{2}\,k+\sqrt{\left(\tfrac{3}{2}\,k \right)^2 + 3\,n} \right) $$ and the intervals $\mathcal{L}_{n,k} = \left]\ell_{n,k}-\ln 3,\ell_{n,k}\right]$. For all $n \geq 1$, define $$…

In this paper we study linear correlations of the divisor function tau(n) = sum_{d|n} 1 using methods developed by Green and Tao. For example, we obtain an asymptotic for sum_{n,d} tau(n) tau(n+d) ... tau(n+ (k-1)d).

Let $A_{\pi}(n,1)$ be the $(n,1)$-th Fourier coefficient of the Hecke-Maass cusp form $\pi$ for $\rm SL_3(\mathbb{Z})$ and $ \omega(x)$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum…

For a function $f\colon \mathbb{N}\to\mathbb{N}$, define $N^{\times}_{f}(x)=\#\{n\leq x: n=kf(k) \mbox{ for some $k$} \}$. Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and…

We consider a binary quadratic variant of the Titchmarsh divisor problem and give an asymptotic formula for $\sum_{p^2+q^2\leq N} \tau(p^2+q^2+1)$, where $p,q$ are primes.

Let $\gcd(j,k)$ be the greatest common divisor of the integers $j$ and $k$. In this paper, we give several interesting asymptotic formulas for weighted averages of the $\gcd$-sum function $f(\gcd(j,k)) $ and the function $\sum_{d|k,…

Let $\zeta^k(s) = \sum_{n=1}^\infty \tau_k(n) n^{-s}, \Re s > 1$. We present three conditional results on the ternary additive correlation sum $$\sum_{n\le X} \tau_3(n) \tau_3(n+h),\quad (h\ge 1),$$ and give numerical verifications of our…

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$…

In a recent paper, Lapkova uses a Tauberian theorem to derive the asymptotic formula for the divisor sum $\sum_{n \leq x} d( n (n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove her result by…

In this short note, we derive an upper-bound for the sum of two comparison functions, namely for the sum of a class K and an extended class K function. To the best of our knowledge, the relations derived in this note have not been…

Let us denote by $\tau(n)$ and $\si(n)$ the number and the sum of the divisors of $n$ and by $\vfi$ Euler's function. We give effective upper bounds for $\frac{n}{\vfi(n)}$ in terms of $\vfi(n)$, and for $\frac{\si(n)}{n}$ in terms of…

Let $3\leqslant k\leqslant9$ be a fixed integer, $p$ be a prime and $d(n)$ denote the Dirichlet divisor function. We use $\Delta(x)$ to denote the error term in the asymptotic formula of the summatory function of $d(n)$. The aim of this…

We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} \Lambda(n) \Lambda(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that…

Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|<n\leq x} f(n) \tau(n-h)$, where $\tau$ denotes the divisor function and…

A natural number $n$ is $y$-smooth if the greatest prime factor of $n$ does not exceed $y$. Let $s_{1}$ and $s_{2}$ are $y$-smooth numbers. We consider sums of smooth squares of the binary Titchmarsh divisor problem and give asymptotic…

We improve upon a result of Steinerberger (2024) by demonstrating that for any fixed $k \in \mathbb{N}$ and sufficiently large $n$, there exist integers $1 \leq a_1, \dots, a_k \leq n$ satisfying: \begin{align*} 0 < \left\| \sum_{j=1}^{k}…