Spectral expansion of certain automorphic functions and its number-theoretical applications

The sums
$$ \sum_{q=1}^\infty\sum_{\substack{t=1\\ t^2+\mathcal D\equiv 0\pmod q}}^q e^{2\pi i\frac{mt}q}h\left(\frac{2\pi m\sqrt \mathcal D}{q}\right),\quad \mathcal D^\frac s2\sum_{n=-\infty}^\infty\sigma_{-s}(n^2+\mathcal D)h\left(\frac{\sqrt{n^2+\mathcal D}}{\sqrt{\mathcal D}}\right), $$
where $\mathcal D>0$ and $\sigma_s(n)=\sum_{d|n}d^s$, а $h$ are represented in terms of spectral characteristics of the automorphic Laplacian for the full modular group. With its help the asymptotic formulae for the sums of the type $\sum_{|n|\leqslant P}\sigma_{-s}(n^2+\mathcal D)$ as $P\to\infty$ are obtained. These formulae generalize the author's earlier result $\sum_{|n|<P}\mathcal T(n^2+\mathcal D)=c_1(\mathcal D)P\log P+c_0(\mathcal D)P+O(P^\frac23\log^\frac23P).$