4 problems

The strong Hardy–Littlewood conjecture. For every $\varepsilon>0$,

Let $\Lambda$ be the von Mangoldt function, define the singular series … and for $h\ne0$ let $\mathfrak S(h)=0$ when $h$ is odd, while for even $h$ let … Set…

Let $h$ be an integer with $h\geq 4$, let $\Lambda$ denote the von Mangoldt function, and let $\mathfrak{S}_h$ denote the Goldbach singular series. Asymptotic Goldbach conjecture.…

Hardy–Littlewood–Elliott conjecture. Then