Let \(\mathbb{U}=\{z\in\mathbb{C}:|z|\leqslant 1\}\) be the unit disc at the complex plane, and let \(f:\mathbb{N}\rightarrow\mathbb{U}\) be a multiplicative function. The author of this paper studies sums \[ \sum\limits_{n\equiv\, a\pmod{q}}f(n), \] where \((a,q)=1\) and \(1\leqslant q\leqslant x\) with \(q\) being very large as a function of \(x\). The results are derived for prime moduli, for moduli in the middle range \(q\leqslant x^{1/2-o(1)}\), for smooth moduli and for general moduli. In addition, hybrid results are also presented. For instance, in the case of of prime moduli, the following assertion is obtained.
Let \(1\leqslant Q\leqslant x/10\) and \(\big(\log(x/Q)\big)^{-1/200}\leqslant\varepsilon\leqslant 1\). Then there exists a set \( [1,x^{\varepsilon^{200}}]\cap\mathbb{Z}\subset Q_{x,\varepsilon}\subset[1,x]\cap\mathbb{Z}\) with \(\big|[1,Q]\setminus Q_{x,\varepsilon}\big|\leqslant (\log x)^{\varepsilon^{-200}}\) such that the following holds.
Suppose \(p\in Q_{x,\varepsilon}\cap [1,Q]\) is a prime, and \(f:\mathbb{N}\rightarrow \mathbb{U}\) is a multiplicative function. If \(\chi_1\) be a character \(\pmod{p}\) minimizing the distance \(\inf_{|t|\leqslant \log x}\mathcal{D}_p(f,\chi(n)n^{\mathrm{i}t},x)\), then \[ \sum^*\limits_{a\,\pmod{p}}\bigg|\sum_{\substack{n\leqslant x}{n\equiv a\pmod{p}}}f(n)-\frac{\chi_1(a)}{\varphi(p)}\sum_{n\leqslant x}f(n)\overline{\chi_1}(n)\bigg|^2\ll_\varepsilon \frac{x^2}{p}, \] where \[ \mathcal{D}_p(f,g,x)=\bigg(\sum_{q\leqslant x, q\neq p}\frac{1-\mathrm{Re}\big(f(q)\overline{g(q)}\big)}{q}\bigg)^{1/2} \] and \(\sum^*_{a\,\pmod{p}}\) denotes a sum over reduced residue classes \(\pmod{p}\).