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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 …

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 …

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 …

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 …

Observe that for any Schwartz function $f \in \mathcal{S}(\mathbb{R})$ having $$ f(0) = \widehat{f}(0) = 1 $$ and $$ f, \widehat{f} \geq 0 \quad \textrm{outside of} \quad [-1,1], $$ the following ri …

Let $-D < 0$ be a negative fundamental discriminant and let $y$ range over the values $y = y_Q = \frac{\sqrt{|D|}}{2a}$, as the values $(a,b,c)$ run through the reduced binary quadratic forms $Q = aX^ …

The elements of $S$ are conjectured to be $\mathbb{Q}$-linearly independent, and so a basis for the $\mathbb{Q}$-linear span of the multiple zeta values. This is what Francis Brown accomplished at t …

In Colmez's formulation, it is necessary that the endomorphism ring be the maximal order $\mathcal{O}_k$. It is then proved only in special cases ($k/\mathbb{Q}$ abelian), or on average over the CM ty …

Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be gener …

Assume without loss of generality that $P$ is irreducible, and denote by $S_P(X)$ the set of primes $p < X$ that divide some value $P(n)$. Let $G$ be the Galois group of $P$ and $n_1 > 0$ the number o …

Here is an elementary proof, the basic idea for which is in Selberg's 1949 paper "An elementary proof of Dirichlet's theorem about primes in an arithmetic progression" (Ann. Math., vol 2, 1949, pp. 29 …

On the other hand, a variant of the question has a positive answer. This question was raised by Chowla in 1964 in the case that $q = p$ is prime and $f(p) = 0$ (but with $f$ taking arbitrary rational …

For $\kappa \in (0,1)$, let $\lambda(\kappa)$ be the density of primes $p$ having $\mathrm{ord}^{\times}_p{2} < \kappa p$. Does $\alpha := \liminf_{\kappa \to 0} \lambda(\kappa)/\kappa > 0$? Is it f …