9 problems

$k$-$\ell$ additive divisor conjecture. There exists a triple $(\vartheta_{k,\ell},C_{k,\ell},\beta_{k,\ell})\in[1/2,1)\times[0,\infty)\times(0,1]$ such that, for every positive ab…

Let $D_{f;4,4}(r)$ denote the smoothed shifted convolution sum formed from the fourfold divisor function, with $f$ smooth and supported in $[X,2X]\times[Y,2Y]$, where $Y\asymp X$,…

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 $X,Y,P\geq1$ with $Y\asymp X$, and let $f$ be smooth, supported in $[X,2X]\times[Y,2Y]$, with the derivative bounds stated in the source. Let … where…

Additive divisor conjecture.

For fixed $k\geqslant2$, write … where $P_{2k-2}(\log x;h)$ is a polynomial of degree $2k-2$ in $\log x$. In particular, let $\Delta_3(x,h)$ denote the error term for $k=3$. Ivić's…

Averaged additive divisor conjecture. For every fixed $k\geqslant 3$, there should exist some $\delta_k>0$ such that, uniformly for

Vinogradov's conjecture. For the relevant, but unspecified, range of $h$, one should have