20 problems

Let $x\geq2$, $q\in\mathbb{N}$, and $a,h\in\mathbb{Z}$ with $(a,q)=1$. Define … where $\mathbf{1}_{(a(a+h),q)=1}$ equals $1$ when both $a$ and $a+h$ are coprime to $q$, and equals…

Let $b(n)=\frac{\phi(n)\Lambda(n)}{n}$, where $\phi$ is Euler's totient function and $\Lambda$ is the Mangoldt function. Let $C(h)$ be the Hardy–Littlewood prime-pair constant, def…

Let … with $\psi_{2}(N,k)$ and $\mathfrak{S}(k)$ as above. Prime Pair Error Lower Bound Conjecture. … The conjecture gives a lower bound of the expected order for the total prime-p…

Let … where $\psi_{2}(N,k)$ counts the von Mangoldt-weighted prime pairs of difference $k$, and $\mathfrak{S}(k)$ is the Hardy–Littlewood singular series. Prime Pair Error Conjectu…

Let $\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field and let $C_1$ be the fixed cone. Let $PQ_M(-d)$ count pairs of primes $(p,q)$ less than $M$ that split completely, with…

Let $E_{d,g}(x)$ be the error contribution defined by … where $S_{k,l}(x;d,g)$ is the sum defined in the source, and let $[d,g]$ be Artin admissible. Artin prime-pair error-term co…

Fix an even integer $d$ and an integer $g\in\mathbb{Z}\setminus\{-1\}$ satisfying the hypotheses in the source, and let $[d,g]$ be the Artin-admissibility condition. Let $\pi_d(x)$…

Let $z$ and $q$ be fixed, let $\mathcal{N}_q(a,b)$ denote a coset lattice congruence class for prime pairs $(p_i,p_{i+z})$, and let $\Psi_z(x;\mathcal{N}_q(a,b),q)$ be the correspo…

Let $z$ and $q$ be fixed, let $\mathcal{N}_q(a,b)$ denote a coset lattice congruence class for prime pairs of gap $z$, and let $\pi_z(x;\mathcal{N}_q(a,b),q)$ count the pairs…

Let $m_1,m_2$ be integers of the same parity. Vatwani's Gaussian prime pair conjecture. There are infinitely many pairs of rational primes $(p_1,p_2)$ of the form … The conjecture…

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…

Let $k$ be even, let $\pi_k(x)$ count primes $p\leq x$ for which $p+k$ is prime, and define … Bateman--Horn prediction for fixed prime gaps. … This refines Polignac's infinitude as…

For $d>0$, define … and write $G(x,d)=\widetilde{G}(x,d)+E(x,d)$. The uniform weak error conjecture for prime-pair counts. The variable-difference Hardy–Littlewood formula holds wi…

For $d>0$, define … and let $\mathfrak{S}(d)$ be the singular series for the pair difference $d$. The variable-difference Hardy–Littlewood conjecture. There is an error term…

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

Sun's conjecture. The least positive integer $m$ for which both sets have cardinality $n$ is the least prime $p\geq 2n-1$ such that $p+2$ is also prime.

Let $(p,q)$ be a pair of primes, and let $g$ denote the relevant Frobenius number and $G_2$ the established upper bound for the associated representation problem. The bound…

For $r\in\mathbb{N}$, define … where $s=\sigma+i\tau$, and let $\rho$ range symmetrically over the nontrivial zeros of the Riemann zeta function. Average Dirichlet-series conjectur…

Let $C_{2r}$ denote the Hardy–Littlewood constant associated with the prime pair $(p,p+2r)$. An arithmetic subsequence of the index sequence $\{2r\}$ means a subsequence obtained b…

For $k\ge 2$ and $r\in\mathbb{Z}\setminus\\{0\\}$, let $C^k_{2r}$ denote the Bateman–Horn constant associated with the prime pair $(p,p^k+2r)$. Mean-value-one conjecture. The const…