21 problems

Let $G(x)$ denote the number of primes in the set $\left\{\left\lfloor x/n\right\rfloor:1\le n\le x\right\}$. Suppose that $x=p_1p_2p_3$, where $p_1,p_2,p_3$ are primes satisfying…

Square-prime short-interval conjecture. For all $x \geqslant 2$,

Asymptotic expansion conjecture. There exists a constant $\kappa>0$ such that

Marques–Trojovský conjecture. The function $\Psi(x)$ satisfies: (i) $\Psi(x)=21$ for all $x\in[0.9,1.3]$; (ii) $\Psi(x)=149$ for all $x\in[0.5,0.8]$; (iii) $\Psi(x)=59{,}875$ for a…

First Hardy–Littlewood conjecture. Unless $P$ forms a complete residue class with respect to some prime, $\pi_P(x)$ is asymptotic to

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

Let $f(t)$ and $g(t)$ be strictly increasing functions tending to infinity as $t$ tends to infinity. Let $\bar D_{f,g}(t)$ count good pairs $(m,n)$ with $m,n$ prime, $m\leq f(t)$,…

Let $\bar D(t)$ count good pairs $(m,n)$ with $m,n$ both prime and $m\leq t$, $n\leq t$, where goodness means that $\Phi_m(\Phi_n(x))$ factors over the integers. Let $\pi(t)$ denot…

Let $x\geq 2$ be large, let $r>0$ be real, and set $y=\log^r x$. With $\pi(t)=\#\{p\leq t:p\text{ is prime}\}$, the oscillation conjecture. Both inequalities … and … occur infinite…

Shifted-index restricted-prime conjecture. For every fixed positive integer $k$, the sequence $(A_n^{(k)})$ satisfies the Restricted Prime Number Theorem. The conjecture is present…

Let $(A_n)$ be defined by $A_n=\sum_{i=1}^{2n}(-1)^ip_i$, and let $r_k=A_m$ be the $k$th prime occurring in this sequence. Alternating-sum prime lower-bound conjecture. For every…

Positive-exponent conjecture. There exists an infimum

Let $\pi(x)$ be the prime-counting function. For a positive integer $c$, let $I_k=[(k-1)c,kc)$ and define $\chi_{I_k}=1$ if $I_k$ contains one or more primes, and $\chi_{I_k}=0$ ot…

Let $m,n\in\mathds{N}$, let $\pi(x)$ denote the prime-counting function, and let $\lceil\cdot\rceil$ denote the ceiling function. Mitra–Paul–Sarkar's conjecture. If … then … This c…

Negative-jump conjecture. Negative jumps of $\eta_N(x)$ occur at every point $x+1\in\mathcal P$, where $\mathcal P$ is the set of primes, and

Normalized Chebyshev-prime criterion. A Chebyshev prime of index $l$ is a prime $p_m$ satisfying the displayed inequality. This criterion is the formulation used to relate generali…

Prime-power midpoint conjecture. The jump satisfies

Midpoint jump conjecture. The jump at the prime $p_n$ satisfies

Let $\pi_b(k)$ denote the number of base $b$ dismal primes with $k$ digits. Prime-counting conjecture. As $k$ tends to infinity, … The conjecture predicts that asymptotically almos…

Let $k$ be a positive integer, and let $n$ be a positive integer. The upper-bound conjecture for primes between $n$ and $kn$. The number of primes between $n$ and $kn$ is at most ……

For a positive integer $n$, let $leg(n)$ denote the number of primes between $n^{2}$ and $(n+1)^{2}$. The improved bound conjecture for $leg(n)$. … The source proposes this as a ti…