In this talk, we will focus on the relations in pro-p extensions of number fields, specially when the root discriminant is bounded. As application, we will give new records on limsup of root discriminants (ie on the constants of Martinet). We will also produce some infinite unramified extensions of number fields where the set of splitting is infinite.

This is a joint work with F. Hajir (UMass) and R. Ramakrishna (Cornell).

During the last years, together with S. Checcoli and F. Veneziano we investigated the Torsion Anomalous Conjecture (TAC). This conjecture received a lot of attention over several years and it is well known to imply the Mordell-Lang Conjecture. Since to prove it in general it remains out of reach for us, we decided to study explicit proofs of special cases and their implications on the Mordell-Conjecture. We were successful in finding the rational points on some new families of curves of growing genus. I will explain our method, its limit and generalisations.

This talk will introduce a variety of mathematics textbooks used in Rome between the 16th and 17th centuries. Through solving mathematical problems together, we will examine the significance of the content via the history of number theory. As long-distance travel from Europe to Asia became routine, mathematics books travelled from Rome to other parts of the world. We will discuss the ways in which Roman intellectual culture influenced the cultures of England, China, and Japan.

The homogeneous form $\Phi_n(X,Y)$ of degree $\varphi(n)$ which is associated with the cyclotomic polynomial $\phi_n(X)$ is dubbed a cyclotomic binary form. A positive integer $m\ge 1$ is said to be representable by a cyclotomic binary form if there exist integers $n,x,y$ with $n\ge 3$ and $\max\{|x|, |y|\}\ge 2$ such that $\Phi_n(x,y)=m$. These definition give rise to a number of questions that we plan to address.

We prove an explicit formula, analogous to the classical explicit formula for $\psi(x)$, for the Cesáro-Riesz mean of any order $k>0$ of the number of representations of $n$ as a sum of two primes. Our approach is based on a double Mellin transform and the analytic continuation of certain functions arising therein.

(joint work with J.Brüudern and J.Kaczorowski)

Speiser in 1935 showed that the Riemann hypothesis is equivalent to the first derivative of the Riemann zeta function having no zeros on the left-half of the critical strip. This result shows that the distribution of zeros of the Riemann zeta functin is related to that of its derivatives. The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates under the truth of the Riemann hypothesis. This result is further improved by Ge. In the first half of this talk, we introduce these results and generalize the result of Akatsuka to higher-order derivatives of the Riemann zeta function. Analogous to the case of the Riemann zeta function, the number of zeros and many other properties of zeros of the derivatives of Dirichlet L-functions associated with primitive Dirichlet characters were studied by Yildirim. In the second-half of this talk, we improve some results shown by Yildirim for the first derivative and show some new results. We also introduce two improved estimates on the distribution of zeros obtained under the truth of the generalized Riemann hypothesis. We also extend the result of Ge to these Dirichlet L-functions when the associated modulo is not small. Finally, we introduce an equivalence condition analogous to that of Speiser's for the generalized Riemann hypothesis, stated in terms of the distribution of zeros of the first derivative of D\ irichlet L-functions associated with primitive Dirichlet characters.

Recall that an integer $n$ is called $y$-smooth when each of its prime divisors is less than or equal to $y$. It is conjectured that, for any $a>0$, any polynomial of positive degree having integral coefficients should possess infinitely many values at integral arguments $n$ that are $n^a$-smooth. One could consider this problem to be morally dual to the cognate problem of establishing that irreducible polynomials assume prime values infinitely often, unless local conditions preclude this possibility. This smooth values conjecture is known to be true in several different ways for linear polynomials, but in general remains unproven for any degree exceeding 1. We will describe some limited progress in the direction of the conjecture, highlighting along the way analogous conclusions for polynomial smoothness. Despite being motivated by a problem in analytic number theory, most of the methods make use of little more than pre-Galois theory. A guest appearance will be made by several hyperelliptic curves.

[This talk is based on work joint with Jonathan Bober, Dan Fretwell and Greg Martin].