25 problems

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 $\psi_2(n)=\sum_{m+m'=n}\Lambda(m)\Lambda(m')$, where $\Lambda$ is the von Mangoldt function. Fujii's formula, under the Riemann Hypothesis, expresses the average Goldbach repr…

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 $R(n)$ 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…

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

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 $n$ as a sum of two primes, and let $\varphi(n)$ be Euler's totient function. For sufficiently large $x$ and $q\leq x/4$, weighte…

Let $R(n)$ denote the number of representations of $n$ as a sum of two primes. For sufficiently large $x$ and $q\leq x/4$, exceptional-interval Goldbach conjecture. There is an abs…

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 $g(n)$ denote the number of representations of an even integer $n$ as a sum of two primes. Fei's weakened Hardy--Littlewood conjecture. There exists a positive constant $delta$…

Landau's average-order conjecture. The function $g(n)$ should be approximated by

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…

Let $X$ tend to infinity through integer values, and let $\Lambda$ be the von Mangoldt function. Let $\mathfrak{S}(X)$ denote the singular series defined in the source. Quantitativ…

Let $n$ be a Gaussian integer with norm $N(n)>10$, and let $p,q$ be Gaussian primes. Conjecture F. One can write $n=p+q$ so that the angles between $n$ and $p$, and between $n$ and…

Let $Q$ denote the relevant set of Gaussian integers, and let $k+ki$ be a Gaussian diagonal integer with $k>1$. A Gaussian diagonal prime is a Gaussian prime of the form $a+ib$ in…

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

Let $N$ be a sufficiently large integer, and let $f_{1},\ldots,f_{r}$ be numbers of the specified type for the representation problem under consideration. Generalized Waring–Goldba…

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

Let … where … and $\Lambda$ is the von Mangoldt function. Natural-boundary conjecture. The line $\Re s=r-1$ would be the natural boundary of $\Phi_r(s)$. This conjecture concerns t…

Let $p_r$ denote the $r$-th prime and let … be the primorial number of order $r$. Let … where $C_2$ is the twin-prime constant, and let $\gamma$ be the Euler–Mascheroni constant. H…

Fujii's conjecture. If

Weighted binary Goldbach conjecture. For any sufficiently large even $n$, the quantity $\mathcal R(n)$ is asymptotically equal to