17 problems

Let $0<\theta<1$ be fixed, let $h\neq0$ be a fixed even integer, and let $\Lambda(n)$, $\mu(n)$ and $\varphi(q)$ denote the von Mangoldt function, Möbius function and Euler totient…

For a squarefree integer $Q$ with $6\mid Q$, let $U_Q$ be the set of residues $r\bmod Q$ such that both $r$ and $r+2$ are coprime to $Q$. Define the residue-pairing bound … where ……

Let $p_n$ denote the $n$th prime. Twin-square-interval conjecture. For every integer $n\geq 3$, there exists at least one pair of twin primes lying in the interval … This is a stro…

Type-III bias conjecture. As $x\rightarrow\infty$,

Type-II bias conjecture. For any two residue classes $a_i$ and $a_j$,

Twin-prime progression conjecture. For every natural number $l$ with $l>2$, there exist arithmetic progressions of twin primes with $l$ terms.

Universal primorial conjecture. There are expected to be infinitely many numbers $x=Kp_n\#+g$ such that both $x$ and $x+2$ are prime.

Lillie's conjecture. The total expected number of primorial twin prime pairs is approximately three.

Wolf's conjecture. The number of sign changes of $d_2(n)$ for $n\in(1,T)$ is $\frac{\sqrt T}{\log T}$. This numerical prediction concerns fluctuations around the Hardy–Littlewood e…

Let $H_a$ be the lexicographically first strictly increasing sequence beginning with $H_a(2)=a$ such that $H_a(n)$ is prime if and only if $n$ is prime. For primes $a>b\geq 3$, wri…

Let $a,q$ be positive integers such that $(a(a+2),q)=1$. Let $\Pi_2$ denote the twin-prime constant, let $\varphi_2(q)$ denote the corresponding arithmetic-progression factor, and…

Twin Ramanujan-prime asymptotic conjecture. If $\pi_2(x)\to\infty$ as $x\to\infty$, then

Twin Ramanujan-prime ratio conjecture. For all $x\ge10^5$,

Let $m\geq4$, set $A_m=\{1,\ldots,m-3\}$, and define … with, for $n\geq2$, … Let $n^*=n^*(m)$ be the point of the last nontrivial increment of $\{\tilde{c}^{(m)}(n)\}$ on $A_m$, wi…

Let $\tilde{c}(n)$ be the sequence defined by … An increment is called main if it occurs at one of the distinguished fundamental points described in the paper. Shevelev's main-incr…

Write … A normalized-increment bound conjecture asserts that, for every $i\geq6$, … The paper uses this proposed bound to derive the stated sufficient condition for infinitely many…

Let $p_n$ denote the $n$th prime, and let its weight be the weight in the paper's decomposition of primes. The smallest member of each twin-prime pair other than $3$ has weight $3$…