We prove an André–Oort-type result for a family of hypersurfaces in ${\mathbb{C}}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel–Tatuzawa lower bound for the class number. We prove that, for $m, n \in {\mathbb{Z}}_{\gt0}$, there exists an effective constant $c(m, n)\gt0$ with the following property: if pairwise distinct singular moduli $x_1, \ldots, x_n$ with respective discriminants $\Delta_1, \ldots, \Delta_n$ are such that $a_1 x_1^m + \cdots + a_n x_n^m \in {\mathbb{Q}}$ for some $a_1, \ldots, a_n \in {\mathbb{Q}} \setminus \{0\}$ and $\# \{ \Delta_i \;:\; {\mathbb{Q}}(\sqrt{\Delta_i}) = K_*\} \leq 1$, then $\max_i \lvert \Delta_i \rvert \leq c(m, n)$. In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples $(x_1, x_2, x_3)$ of singular moduli such that $a_1 x_1 + a_2 x_2 + a_3 x_3 \in {\mathbb{Q}}$ for some $a_1, a_2, a_3 \in {\mathbb{Q}} \setminus \{0\}$.

Turán observed that logarithmic partial sums $\sum_{n\le x}{f(n)}/{n}$ of completely multiplicative functions (in the particular case of the Liouville function $f(n)=\lambda(n)$) tend to be positive. We develop a general approach to prove two results aiming to explain this phenomena.

Firstly, we show that there exist constants $C, x_0\ge 1,$ such that for any completely multiplicative function f satisfying $-1\le f(n)\le 1$, we have

\begin{equation*}\sum_{n\le x}\frac{f(n)}{n}\ge -\frac{C(\log\log\log{x})^2}{(\log\log{x})}, \quad x\ge x_0.\end{equation*}

$f:\mathbb{N}\to \{-1,1\}$

$\sum_{n\le x}{f(n)}/{n}$

$O(\exp(-\exp({\log x\cdot \log\log\log x}/{C\log \log x}))).$

We obtain $\Omega$-results for linear exponential sums with rational additive twists of small prime denominators weighted by Hecke eigenvalues of Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. In particular, our $\Omega$-results match the expected conjectural upper bounds when the denominator of the twist is sufficiently small compared to the length of the sum. Non-trivial $\Omega$-results for sums over short segments are also obtained. Along the way we produce lower bounds for mean squares of the exponential sums in question and also improve the best known upper bound for these sums in some ranges of parameters.

Let $S_g$ denote the genus g closed orientable surface. A coherent filling pair of simple closed curves, $(\alpha,\beta)$ in $S_g$, is a filling pair that has its geometric intersection number equal to the absolute value of its algebraic intersection number. A minimally intersecting filling pair, $(\alpha,\beta)$ in $S_g$, is one whose intersection number is the minimal among all filling pairs of $S_g$. In this paper, we give a simple geometric procedure for constructing minimally intersecting coherent filling pairs on $S_g, \ g \geq 3,$ from the starting point of a coherent filling pair of curves on a torus. Coherent filling pairs have a natural correspondence to square-tiled surfaces, or origamis, and we discuss the origami obtained from the construction.