8 problems

Let $a$ be a complex parameter and define the generalized divisor function … For the partial sums, write … where $E_a(x)$ is the error term. Extended divisor problem conjecture. Fo…

For each $k\geqslant2$, let $\beta_k$ be the equal-sums exponent for logarithmic random sets, and let $\alpha_k$ be the supremum of the real numbers $\alpha$ such that almost every…

For each $k\geqslant 2$, let $\alpha_k$ be the supremum of the real numbers $\alpha$ such that, for almost every $n\in\mathbb N$, $n$ has divisors $d_1<\cdots<d_k$ with…

Kowalski's conjecture. As $x\to\infty$,

Let $k$ be a positive integer, let $d_k(n)$ be the $k$-fold divisor function, … and let $f$ be a smooth function on $\mathbb{R}^{+}\times\mathbb{R}^{+}$. Vinogradov's generalized s…

Let $d(n)$ denote the divisor function, let $X>0$ and let $h$ be a shift with $h\leq X^{1/2}$. The expected main term has the form $XP(\log X)$, where $P$ is a quadratic polynomial…

Mean-square shifted-product conjecture. The displayed estimate should hold.

Let $1\leq a<b$ be fixed coprime integers, define … and set $D(a,b;x)=\sum_{n\leq x}d(a,b;n)$. Define … Here $varepsilon>0$ is arbitrary. General two-dimensional divisor problem co…