11 problems

Let $\lambda$ be the Liouville function. Fix $k\geqslant1$ and distinct positive integers $h_1,\dots,h_k$. Logarithmic Chowla conjecture. One has … as $X\to\infty$. This logarithmi…

Let $k,l\geqslant 2$ be natural numbers, let $h\in\mathbb{Z}\setminus\{0\}$, and let $d_k(n)$ and $d_l(n)$ denote the $k$-fold and $l$-fold divisor functions. Divisor correlation c…

Elliott-type correlation conjecture. If $P_1$ and $P_2$ are irreducible binary quadratic forms that are not multiples of each other, then they are good for vanishing of correlation…

Let $f$ be a fixed primitive cusp form, holomorphic or Maass, and let $\lambda_f(n)$ denote its Hecke eigenvalue at $n$. Write $\mathcal{L}=\log X$. Integer correlation conjecture.…

Let $f$ be a fixed primitive cusp form, holomorphic or Maass, and let $\lambda_f(p)$ denote its Hecke eigenvalue at a prime $p$. Write $\mathcal{L}=\log X$. Prime correlation conje…

Elliott's conjecture. The limit $L$ exists and is equal to zero unless every $g_j$ pretends to be a twisted Dirichlet character in the displayed sense.

Divisor correlation conjecture. For some polynomial $P_{k,l,h}$ in $\log X$ of degree $k+l-2$, one has, as $X\to\infty$,

For a continuous TASEP on a ring with $n$ particles, let … be the stationary probability that an adjacent pair has values $i$ and $j$. The full correlation conjecture. For $n\ge 2$…

For a continuous TASEP on a ring with $n$ particles, let … be the stationary two-point correlation for an adjacent pair, where $a$ is arbitrary. The upper-right correlation conject…

For a fixed positive integer $h$, define … where $\tau$ is the divisor function. Triple-divisor correlation conjecture. It is conjectured that … as $X\rightarrow\infty$, for a suit…

Consider a nonequilibrium stationary state of a system with non-equilibrium particle-conserving dynamics in dimension $d\geq 2$. Suppose that its correlations between occupation nu…