19 problems

Let $\pi(t)$ denote the number of primes at most $t$, and let $x,y\geq 2$. Second Hardy–Littlewood conjecture. … This is a classical subadditivity conjecture for the prime-counting…

Let $p_n$ denote the $n$th prime, and define … Let $\mathcal{C}_{\mathcal{A}}$ denote the subring of algebraic elements of $\mathcal{A}$. Transcendence conjecture. … The paper prov…

Let $\pi(x)$ denote the number of primes at most $x$, and let $x,y\in\funmathbb{Z}$ satisfy $x,y\geqslant 2$. Hardy–Littlewood's second conjecture. The prime-counting function is s…

Truncated logarithmic-integral bounds conjecture. The inequalities

Truncated asymptotic bounds conjecture. For every real $x\ge e$,

The $li_0$-$li_1$ bounds conjecture. For every real $x\ge e$,

Dirichlet's conjecture. A better approximation to $\pi(x)$ is the offset logarithmic integral function $\operatorname{Li}(x)$, which captures the notion of a prime-number density f…

Logarithmic weighted sum conjecture. One has

Let $\pi(t)$ denote the number of primes at most $t$, and let $p,q>1$ satisfy … There exists a sufficiently large natural number $n_0$ such that, for all $x,y\geq n_0$, Young-type…

Minculete's conjecture.

Let $count$?

The conjectures. (A) For every integer $k\ge0$, the set $\{n\ge1:a_n=k\}$ has a positive density $\delta_k$, and

For $n\in\mathbb{Z}^+$, let $p_n$ denote the $n$th prime and let $\pi(x)$ denote the number of primes not exceeding $x$. Sun's conjecture. For every $n\in\mathbb{Z}^+$, there exist…

For an integer $n>1$, let $\pi(x)$ denote the number of primes not exceeding $x$. Sun's conjecture. For every integer $n>1$, there exists a positive integer $k\le n$ such that … Su…

Let $\beta(t)$ be the function defined in the paper, and let $\mu$ and $\alpha$ be the constants used there. Unimodality conjecture for $\beta$. There exists a constant…

Let $\mathcal{R}\subset \mathbb{R}_+$ be the Ramanujan set of values $x\in \mathbb{R}_+$ for which $\pi(x)$ is the nearest integer to … where $\pi(x)$ is the prime-counting functio…

Let $\pi(x)$ denote the number of primes not exceeding $x$, and let $x$ range over the natural numbers. The three-to-two prime-counting conjecture. One should have … for every natu…

Let $m,n\in\mathbb{Z}^+$, let $\pi(x)$ denote the number of primes not exceeding $x$, and let $p_r$ denote the $r$-th prime. Divisibility conjecture. For every positive integer $m$…

Let $n>1$ be an integer, and let $\pi(x)$ denote the number of primes not exceeding $x$. Prime value conjecture. For some integer $k$ with $1\leq k\leq n$, the number $\pi(kn)$ is…