23 problems

Let $k\ge2$, let $\lambda$ be sufficiently large that the mode is nonzero, and let $m_k(\lambda)$ denote a mode of the Poisson distribution of order $k$, with mean parameter…

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…

The $9/(4k-1)$ bound conjecture. The pmf decreases monotonically for all $n\geq k$ whenever

The exact-threshold conjecture. For fixed $k\geq 2$,

Let $k\ge2$, let $\lambda$ be sufficiently large that the mode is nonzero, and let $m_k(\lambda)$ denote a mode of the Poisson distribution of order $k$, with mean parameter…

Let $k\ge2$ be fixed, let $\lambda\in(\frac{1}{k}\ln2,1)$, and write $\kappa=\frac{k(k+1)}{2}$. Let $\nu_k(\lambda)$ denote the median of the Poisson distribution of order $k$. Con…

For fixed $k\ge 2$ and $\lambda>0$, let $m_{k,\lambda}$ denote the mode of the Poisson distribution of order $k$. The constants $c_0$ and $c_1$ are to be determined. Improved lower…

Poisson order-$k$ mode conjecture. Numerical evidence suggests that, for all tested $k\ge1$ and $n\ge2\kappa$,

Asymptotic median-difference conjecture.

Median offset limit conjecture.

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 order-$k$ median conjecture. Numerical evidence suggests that, for all $k\ge1$ and $n\ge\kappa$,

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 $N_{\lambda}\sim \mathrm{Poisson}(\lambda)$, and write $\mathrm{Median}(N_{\lambda})$ for a median of this random variable. Choi's sharp bounds conjecture. … Moreover, these bo…

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

Ramanujan's asymptotic conjecture. For an auxiliary quantity $calpha_b$, one has

Let $q$ be fixed, let $g$ be the genus of a curve over the finite field $\mathbb{F}_q$, and let $n$ be a possible number of rational points. Consider the fraction of curves of genu…

Let $\Pi$ follow a Poisson distribution with mean $\nu$, let $U$ be uniformly distributed on $[0,1]$, and set $Z=\Pi+U$, corresponding to the choice $\varphi(t)=t$. Write…

Let $f_k(x;\lambda)$ denote the probability mass function of the Poisson distribution of order $k$, and let $m_{k,\lambda}$ be the mode or modes of $f_k(x;\lambda)$, namely the val…