13 problems

Let $G_2(n)=\sum_{k+m=n}\Lambda(k)\Lambda(m)$ for positive integers $n$, where $\Lambda$ is the von Mangoldt function, and define … Under the Riemann hypothesis, $\Phi_2(s)$ admits…

Let $r_2'(N)$ denote the number of representations of $N$ as a sum of two primes, and let $\mathfrak{S}(N)$ be the singular series … For even integers $N$, let $r_2(N)$ denote the…

Let $\mathscr P$ denote the set of all primes. For sets $S,T$ of positive integers, define $S\sim T$ to mean that their symmetric difference is finite. Ostmann's inverse Goldbach c…

Let $0<\theta<1$ be fixed, let $N$ be a sufficiently large natural number, and let $\Lambda(n)$, $\mu(n)$ and $\varphi(q)$ denote the von Mangoldt function, Möbius function and Eul…

Define the weighted Goldbach representation function … and the singular series … where $C_2$ is the twin prime constant. Friedlander–Goldston–Iwaniec–Suriajaya conjecture. There ex…

Let $R(n)$ denote the number of representations of an integer $n$ as a sum of two primes, and let $\varphi$ denote Euler's totient function. Jia's strengthened Hardy–Littlewood con…

Let $R(n)$ denote the number of representations of an integer $n$ as a sum of two primes. Weak Hardy–Littlewood conjecture. There exists an absolute constant $c_3>0$ such that, for…

Let $g(n)$ denote the number of representations of an even integer $n$ as a sum of two primes. Exceptional weak Hardy--Littlewood conjecture. Suppose that $x$ is sufficiently large…

Let $N$ be a positive integer, let $\rho_1$ and $\rho_2$ range over the nontrivial zeros of the Riemann zeta function, and let $B_x(a,b)$ denote the incomplete beta function. The n…

Inverse binary Goldbach conjecture. For $N$ sufficiently large depending on $\delta$, the sumset $A_1+A_2$ contains a composite number.

Let $\delta>0$. Dense-set inverse Goldbach conjecture. If $X$ is sufficiently large in terms of $\delta$ and $A,B\subset[X]$ satisfy … then $A+B$ contains a composite number. This…

Conjectured unification. The estimate

Egami–Matsumoto's effective independence conjecture. There exists some $\alpha<\frac{\pi}{2}$ such that either $\{\gamma_1,\gamma_2\}=\{\gamma_3,\gamma_4\}$, or