9 problems

Let $B(k,\mu)=\sum_i {k\brace i}\mu^i$ denote the Bell polynomial, and write $B_x=B(x,1)$. For a Poisson random variable with mean $\mu$, its $k$th raw moment is $B(k,\mu)$. Bell-n…

Let $\pi(t)$ denote the number of primes at most $t$. For a real parameter $\lambda=(\log x)^{o(1)}$ and an integer $k\ll (\log_2 x)^2$, count integers $n\leqslant x$ according to…

Let $f\in\mathbb{Z}[x,y]$ and, for each prime $p$, let $C_p$ be the curve of points modulo $p$ defined by $f(x,y)\equiv 0\pmod p$. For a $k$-tuple…

The implied-Poisson-parameter conjecture. The sequence $\{\lambda_k\}$ is decreasing. This is proposed as a strengthening of the paper's earlier conjecture on the probabilities of…

Median offset monotonicity conjecture. The function $\alpha_{k,n}-n$ is decreasing in $n$ under one of the following conditions: (i) $k=2$ and $n\ge3\kappa$; (ii) $k\in[3,6]$ and…

Median offset bound conjecture. For fixed $k>1$ and $n\ge\kappa$,

Exactness conjecture for the median asymptotics. The terms in these two formulas that depend only on $k$ are exact.

Poisson conjecture. As $x\to\infty$,

Poisson $(S_2)$ conjecture. $X$ satisfies Condition $(S_2)$.