1. Introduction. In this article, we prove a generalisation of the Mertens theorem for prime numbers to number fields and algebraic varieties over finite fields, paying attention to the genus of the field (or the Betti numbers of the variety), in order to make it tend to infinity and thus to point out the link between it and the famous Brauer-Siegel theorem. Using this we deduce an explicit version of the generalised Brauer-Siegel theorem under GRH, and a unified proof of this theorem for asymptotically exact families of almost normal number fields.The classical Brauer-Siegel theorem describes the asymptotic behaviour of the quantity hR (the product of the class number and the regulator) in a family of number fields with growing genus under the conditions that the genus grows much faster than the degree and assuming some additional properties like normality or the Generalised Riemann Hypothesis (GRH) to deal with the Siegel zeroes. These two hypotheses are of different nature: omitting the first changes the final result, while the second is a technical hypothesis. Tsfasman and Vlȃduţ [9] were able to remove the first hypothesis, which led to the so called generalised Brauer-Siegel theorem, and Zykin [10] was able to replace "normality" by "almost normality" in the second hypothesis by using results of Stark and Louboutin. He also managed to generalise the Brauer-Siegel theorem to the case of smooth absolutely irreducible projective varieties over finite fields.As for the Mertens theorem, proven by Mertens in the case of Q, and much later generalised by Rosen [5] to both number and function fields, it can be regarded as the Brauer-Siegel theorem in the global field or variety constituting the family. An explicit Mertens theorem leads therefore to an explicit formulation of the generalised Brauer-Siegel theorem. We first recall the formulations of the (generalised) Brauer-Siegel theorem and Mertens theorem, then we prove their explicit versions for number fields and smooth

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Abstract. In this article we study certain asymptotic properties of global fields. We consider the set of Tsfasman-Vlȃduţ invariants of infinite global fields and answer some natural questions arising from their work. In particular, we prove the existence of infinite global fields having finitely many strictly positive invariants at given places, and the existence of infinite number fields with certain prescribed invariants being zero. We also give precisions on the deficiency of infinite global fields and on the primes decomposition in those fields.

Let f be a primitive cusp form of weight k and level N, let χ be a Dirichlet character of conductor coprime with N, and let L(f ⊗ χ, s) denote either log L(f ⊗ χ, s) or (L ′ /L)(f ⊗ χ, s). In this article we study the distribution of the values of L when either χ or f vary. First, for a quasi-character ψ : C → C × we find the limit for the average Avg χ ψ(L(f ⊗ χ, s)), when f is fixed and χ varies through the set of characters with prime conductor that tends to infinity. Second, we prove an equidistribution result for the values of L(f ⊗ χ, s) by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average Avg h f ψ(L(f, s)), when f runs through the set of primitive cusp forms of given weight k and level N → ∞. Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for L(f ⊗ χ, s).