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Abstract: Let V be an orbit in of a finitely generated subgroup Λ of whose Zariski closure is suitably large (e.g. isomorphic to ). We develop a Brun combinatorial sieve for estimating the number of points on V for which a fixed set of integral polynomials take prime or almost prime values. A crucial role is played by the expansion property of the ‘congruence graphs’ that we associate with V. This expansion property is established when . To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

Let be an L-function which appears in the Langlands–Shahidi theory. We give a lower bound for when using Eisenstein series. This method is applicable even when is not known to be absolutely convergent for . To cite this article: S.S. Gelbart et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004). [Copyright &y& Elsevier]

Abstract. We present a new method for establishing the "gap" property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S[sup 2] as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[SU(2)], the "unfolded" consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N arrow right Infinity; we establish several results in this direction. For certain special "arithmetic" (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction. [ABSTRACT FROM AUTHOR]