9 problems

Let $d_k(n)$ denote the number of ordered factorizations of $n$ into $k$ positive integers, and let $d_l(n)$ be defined similarly. Let $\boldsymbol{\epsilon>0}$ and $k,l\ge 2$. For…

Let $k,\ell\in\mathbb{N}$ and let $\mathcal{AD}_{k,\ell}(\vartheta_{k,\ell},C_{k,\ell},\beta_{k,\ell})$ denote the averaged $k$–$\ell$ additive divisor hypothesis stated in the pap…

Let $\mathcal{I}=\{a_1,\ldots,a_k\}$ and $\mathcal{J}=\{b_1,\ldots,b_\ell\}$ be sets of distinct complex numbers with $|a_i|,|b_j|\ll(\log X)^{-1}$. Let…

Let $k,\ell>1$ be natural numbers, let $x$ be large, and let $h$ satisfy $1\le h\le x^{1-\varepsilon}$. Let $D_{k,\ell}(x,h)=\sum_{n\le x}\tau_k(n)\tau_\ell(n+h)$, and let…

Let $k$ and $cell$ be integers with $k,cell\ge 2$, let $\varepsilon>0$, and define … where $\tau_k$ is the $k$-th divisor function. Additive divisor conjecture, simplified version.…

Let $U(z)=y^{k/2}f(z)$, where $f$ is a holomorphic Hecke cusp form of weight $k$, with Fourier coefficients $\lambda(n)$ and first coefficient $\rho(1)$. For fixed $0<a\le y\le b$…

Let $U(z)$ be a Hecke–Maass cusp form with Laplace eigenvalue $1/4+T^2$, Fourier coefficients $\lambda(n)$ and first coefficient $\rho(1)$. For fixed $0<a\le y\le b$,…

Let $U(z)$ be a Hecke–Maass cusp form with Laplace eigenvalue $1/4+T^2$, Fourier coefficients $\lambda(n)$, and first Fourier coefficient $\rho(1)$. For fixed $0<a\le y\le b$, nonz…

Let $t(n)$ denote the Fourier coefficients of the non-holomorphic cusp form considered in the paper, and let $m\sim M$ mean $M<m\leq 2M$. Let $N\geq 1$, $1\leq L\leq N$, and…