18 problems

For each $\ell\in\mathbb{N}$, let $M_{2\ell}(n,1)$ denote the arithmetic function defined in the paper. Asymptotic conjecture. There exists $\beta_{\ell}\in\mathbb{Q}$ such that ……

Let $f$ be any small arithmetic function, and let $S_f(x)$ and $C_f$ be as in the paper. The $2/5$ error-term conjecture. One has … The preceding mean-square theorem shows that the…

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…

Chowla–Todd's conjecture.

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…

For $k\ge2$, let … where $p_1,\ldots,p_k$ range over primes and $[\,\cdot\,]$ denotes the least common multiple. Let $\mu_k>0$ be the constant in the conjectured asymptotic for…

For every $k\ge2$, define the $k$th moment of the shifted-prime divisor function by … and let $\mu_k>0$ be a constant depending on $k$. Higher-moment conjecture. One may conjecture…

Let $q$ be a positive squarefree integer, and let $D_{2,q}(N)$ denote the number of $D(q)$-pairs whose elements lie in $\{1,2,\ldots,N\}$. For a positive integer $n$, let $h(n)$ de…

Let $\tilde{N}_{2}(x)$ denote the number of positive integer primitive triples $(i,j,k)$ with $i+j=k\leq x$ and $i,j,k$ square-full. Brown and Vaaler’s conjecture. … as…

For $x\geq 1$, let the base-10 SanD numbers below $x$ be the ordered pairs $(a,b)$ satisfying $1\leq a<b\leq x$ and $b-a=s_{10}(ab)$, where $s_{10}$ is the decimal digital-sum func…

A base-10 SanD number is an ordered pair $(a,b)$ with $1\leq a<b$ satisfying $b-a=s_{10}(ab)$, and a SanD prime is a SanD number $(p,q)$ in which both entries are prime. Here…

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

Extension conjecture. This asymptotic should hold for all fixed $z>0$ in place of $m\in\mathbb{N}$.

For a positive square-free integer $d$, let $K=\mathbb Q(\sqrt d)$ be the corresponding real quadratic field, and let $h(d)$ denote its class number. Let $h(k,X)$ be the number of…

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 $x\to\infty$,

Dirichlet divisor moment conjecture. There are an explicit constant $C_k$ and some $c(k)>0$ such that