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By a "globally bounded $G$-function," following G. Christol, I will mean a solution to a (minimal) linear differential equation on $\mathbb{P}^1$ (then necessarily of the Fuchsian type and with ration …

Duke, and Linnik before him under a restrictive condition, proved that the set of closed geodesics of a given length $L$ is equidistributed on the modular surface as $L \to \infty$. This is a theorem …

Gleason's polynomials are the sequence of monic integer polynomials defined recursively by $$ \prod_{d \mid n} G_d(c) = (((c^2+c)^2+c)^2+\cdots+c)^2+c \quad \quad \quad [\textrm{$n$ iterates}], $$ for …

My question concerns holonomic sequences of rational numbers, meaning assignments $a(n) : \mathbb{N} \to \mathbb{Q}$ fulfilling a linear recurrent relation of the form $$ a(n+k) = \sum_{i=0}^{k-1} p_i …

Tautologically, the integer polynomials form a discrete set in $L^1$ of the unit circle. On the other hand, a set of logarithms ordered by norm becomes generally rather denser than the original set. …

In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree …

Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and $$ c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^ …

Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are fi …

Here is another possible refinement of the Lehmer conjecture. For $\alpha \in \overline{\mathbb{Q}}^{\times}$, let $C_{\alpha} \subseteq \mathbb{Q}(\alpha)$ be the maximal cyclotomic field contained …

As $N \to \infty$, the set of fractional parts $$ \Big\{ \frac{N}{n}\Big\}, \quad 1 \leq n < \sqrt{N} $$ becomes asymptotically equidistributed in the Lebesgue measure of $[0,1]$. The same persists w …

The question. Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$? The rather cr …

Consider integer polynomials $P \in \mathbb{Z}[X] \setminus \{0\}$ of a degree $D \geq 1$ and without multiple complex roots. Let me introduce a notation $$ d(P) := \frac{1}{D} \log{|\mathrm{Disc}(P)| …

This is known, and follows from Theorem 2 in Harcos and Michel's paper The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points. II (Invent. math., vol. 163, 20 …

Since you ask more generally for results on $E(\mathbb{Q}^{\mathrm{ab}})$, let me expand my comment into a short answer. Amoroso and Dvornicich discovered (A lower bound on the height in abelian ext …

Because the scale is too small in Mertens's theorem, and the prime number theorem as well as the Riemann hypothesis are hidden by the $O(1/\log{X})$ notation. Indeed, the former amounts to strengthen …