3 problems

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…