7 problems

Let $\mathcal H(n,1)$ denote the Hankel determinant studied in the paper. For each fixed integer $k\geq 0$, consider the values with index $n=2^m+k$. Asymptotic product formula con…

Let $p_m$ denote the $m$-th prime, set $n_m=p_{m-1}+1$, and let $L_m$ be the largest prime factor of $n_m$ when $L_m>m$; define $R_m=n_m/L_m$. For a fixed positive integer $r$, wri…

Let $n=n(m)$, let $\Gamma_m\subset\mathrm{GL}^+_2(\mathbb{R})^n$ be a sequence of groups commensurable with a Hilbert modular group, and suppose that whenever $m\ne m'$ and…

Let $p_n$ denote the $n$th prime, and let $S(n)$ denote the Shapiro-class function. Let $B=\gamma/\log 2$, where $\gamma$ is the Euler–Mascheroni constant. Then Shapiro-class formu…

Shapiro-class formula for $\pi(n)$. Then $G(n)\sim\pi(n)$ for a constant $b$, where

For each positive integer $n>2$, define … Here $p_n$ denotes the $n$th prime number. Asymptotic formula for the nth prime. There exist some $\epsilon\in[0,1]$ and $\delta\in[-2,2]$…

Short-average prime-pair conjecture. As $x\to\infty$,