14 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 … 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…

A Sophie Germain prime is a prime $p$ for which $2p+1$ is also prime, and let $C_2$ be the twin prime constant. Sophie Germain prime conjecture. The number of Sophie Germain primes…

Let $k$ be a positive integer and let $C_{2,k}$ be the prime-pair constant. Extended Goldbach conjecture. The number of ways of writing $2k$ as a sum of two primes is asymptotic to…

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…

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…

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

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…