### A lower bound for the zeros of Riemann's zeta function on the critical line

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We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods in use for computing an equation for the class fields themselves: Kummer theory, Stark units and complex multiplication. Using these techniques we can construct many new number fields, including fields of very small root discriminant.

In this survey article we start from the famous Furstenberg theorem on non-lacunary semigroups of integers, and next we present its generalizations and some related results.

2000 Mathematics Subject Classification: 11D75, 11D85, 11L20, 11N05, 11N35, 11N36, 11P05, 11P32, 11P55.The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.

The Littlewood conjecture in Diophantine approximation claims that$$\underset{q\ge 1}{inf}\phantom{\rule{0.166667em}{0ex}}q\xb7\parallel q\alpha \parallel \xb7\parallel q\beta \parallel =0$$holds for all real numbers $\alpha $ and $\beta $, where $\parallel \xb7\parallel $ denotes the distance to the nearest integer. Its $p$-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that$$\underset{q\ge 1}{inf}\phantom{\rule{0.166667em}{0ex}}q\xb7\parallel q\alpha \parallel \xb7{\left|q\right|}_{p}=0$$holds for every real number $\alpha $ and every prime number $p$, where ${|\xb7|}_{p}$ denotes the $p$-adic absolute value normalized by ${\left|p\right|}_{p}={p}^{-1}$. We survey the known results on these conjectures and highlight recent developments.

Let K = Q(ζp) and let hp be its class number. Kummer showed that p divides hp if and only if p divides the numerator of some Bernoulli number. In this expository note we discuss the generalizations of this type of criterion to totally real fields and quadratic imaginary fields.

A bibliography of recent papers on lower bounds for discriminants of number fields and related topics is presented. Some of the main methods, results, and open problems are discussed.

For each transitive permutation group $G$ on $n$ letters with $n\le 4$, we give without proof results, conjectures, and numerical computations on discriminants of number fields $L$ of degree $n$ over $\mathbb{Q}$ such that the Galois group of the Galois closure of $L$ is isomorphic to $G$.

These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions:(i)Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem?(ii)Is it useful to combine this approach with traditional approaches to Diophantine equations: Diophantine approximation, arithmetic geometry, ...?