5 problems

Let $\mathbb{P}$ be the set of primes. Weak uniform Hardy–Littlewood conjecture. There exists a fixed $\delta>0$ such that, uniformly for every even integer $h\leq x^\delta$ and al…

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…

For $k\geq1$, let $\mathcal{H}=\{h_1,\dots,h_k\}\subseteq\mathbb{Z}$, let $q\equiv1\,(\mathrm{mod}\,4)$ be prime, let $a\in\mathbb{Z}$, and let $R_k(\mathcal{H};x,q,a)$ and…

For $k\geq 1$, let $\mathcal{H}=\{h_1,\dots,h_k\}\subseteq\mathbb{Z}$, let $q\equiv1\,(\mathrm{mod}\,4)$ be prime, let $a\in\mathbb{Z}$, and let $R_k(\mathcal{H};x,q,a)$ be the nor…

Let $a,b,c$ be relatively prime integers with $a>0$, with $a+b$ and $c$ not both even, and with $b^2-4ac$ not a perfect square. Set $f(t)=at^2+bt+c$. Hardy--Littlewood's quadratic…