for any e > 0 and A > 0, the implied constant in the symbol <g depending at most on E and A (see [1] and [14]). The original proofs of Bombieri and Vinogradov were greatly simplified by P. X. Gallagher [4]. An elegant proof has been given recently by R. C. Vaughan [13]. For other references see H. L. Montgomery [10] and H. -E. Richert [12]. Estimates of type (1) are required in various applicat...

We study probability inequalities leading to tail estimates in a general semigroup G with a translation-invariant metric dG . Using recent work [Ann. Prob., to appear] that extends the Hoffmann-Jørgensen inequality to all metric semigroups, we obtain tail estimates and approximate bounds for sums of independent semigroup-valued random variables, their moments, and decreasing rearrangements. In ...

We consider a vector kinetic BGK equation leading to isentropic gas dynamics in the relaxation limit. We prove the existence of a weak solution satisfying a kinetic entropy inequality for any initial data with nite energy. Key-words: isentropic gas dynamics { kinetic BGK model { kinetic entropy Mathematics Subject Classi cation: 35L65, 76N15, 35B35, 82C40 Work partially supported by European TM...