By Duke W.

Articles during this quantity are in keeping with talks given on the Gauss-Dirichlet convention held in Gottingen on June 20, 2005. The convention venerated the a hundred and fiftieth anniversary of the dying of C.-F. Gauss and the two hundredth anniversary of the beginning of J.-L. Dirichlet. the quantity starts off with a definitive precis of the lifestyles and paintings of Dirichlet and keeps with 13 papers through top specialists on study issues of present curiosity in quantity idea that have been at once encouraged by way of Gauss and Dirichlet. one of the issues are the distribution of primes (long mathematics progressions of primes and small gaps among primes), classification teams of binary quadratic varieties, a variety of points of the speculation of $L$-functions, the speculation of modular kinds, and the learn of rational and quintessential recommendations to polynomial equations in numerous variables.

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Reine Angew. Math. : Uber Gauß’ zahlentheoretische Arbeiten. Materialien f¨ ur eine wissenschaftliche Biographie von Gauß, ed. by F. Klein and M. Brendel, Heft 1. : Johann Peter Gustav Lejeune Dirichlet. Dokumente f¨ ur sein Leben und Wirken. (Abh. Dt. Akad. Wiss. Berlin, Kl. , Phys. Techn. 1959, No. : Uber die F¨ orderung deutscher Mathematiker durch Alexander von Humboldt. In: Alexander von Humboldt. Gedenkschrift zur 100. Wiederkehr seines Todestages. Berlin: Akademie-Verlag, 1959, pp. : Dirichletiana.

Xn ) = 1. Throughout this work we shall work with the height metrized by the choice of norm |x| := max0 i n |xi |. Given a suitable Zariski open subset U ⊆ V , the goal is then to study the quantity (1) NU,H (B) := #{x ∈ U (Q) : H(x) B}, as B → ∞. It is natural to question whether the asymptotic behaviour of NU,H (B) can be related to the geometry of V , for suitable open subsets U ⊆ V . Around 1989 Manin initiated a program to do exactly this for varieties with ample anticanonical divisor [FMT89].

Abh. Dt. Akad. Wiss. Berlin, Kl. , Techn. 1960, No. : Alexander von Humboldts wissenschaftsorganisatorisches Programm bei ¨ der Ubersiedlung nach Berlin. -Ber. Dt. Akad. Wiss. -R. ): Briefwechsel zwischen Alexander von Humboldt und Carl Friedrich Gauß. -R. ): Briefwechsel zwischen Alexander von Humboldt und Peter Gustav Lejeune Dirichlet. : Die Mathematik und ihre Dozenten an der Berliner Universit¨ at, 1810– 1933. -R. ): Carl Friedrich Gauß. Der “F¨ urst der Mathematiker” in Briefen und Gespr¨ achen.