By William Duke, Yuri Tschinkel

Articles during this quantity are in accordance with talks given on the Gauss-Dirichlet convention held in GÃ¶ttingen on June 20-24, 2005. The convention venerated the a hundred and fiftieth anniversary of the loss of life of C.-F. Gauss and the two hundredth anniversary of the start of J.-L. Dirichlet. the quantity starts off with a definitive precis of the existence and paintings of Dirichlet and keeps with 13 papers by means of major specialists on study issues of present curiosity in quantity idea that have been at once encouraged by means of Gauss and Dirichlet. one of the subject matters are the distribution of primes (long mathematics progressions of primes and small gaps among primes), type teams of binary quadratic types, a number of facets of the idea of $L$-functions, the idea of modular varieties, and the examine of rational and imperative suggestions to polynomial equations in numerous variables. Titles during this sequence are co-published with the Clay arithmetic Institute (Cambridge, MA).

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**Sample text**

It was probably on this occasion that they made each other’s acquaintance, which soon developed into a devoted friendship. After his return to Berlin, Dirichlet saw to it that Liouville was elected a corresponding member6 of the Academy of Sciences in Berlin, and he sent a letter to Liouville suggesting that they should enter into a scientiﬁc correspondence ([L¨ u], p. ). Liouville willingly agreed; part of the correspondence was published later ([T]). Moreover, during the following years, Liouville saw to it that French translations of many of Dirichlet’s papers were published in his journal.

Associated with each form f is its group of automorphs containing all matrices αβ ∈ SL2 (Z) transforming f into itself. The really interesting quantity now is γ δ the number R(n, f ) of representations of n by f which are inequivalent with respect to the natural action of the group of automorphs. Then R(n, f ) turns out to be ﬁnite, but still there is no simple formula for this quantity. Deﬁne now f to be primitive if (a, b, c) = 1. Forms equivalent to primitive ones are primitive. Denote by f1 , .

1], p. 318). The crucial new tools enabling Dirichlet to prove his theorem are the L-series which nowadays bear his name. In the original work these series were introduced by means of suitable primitive roots and roots of unity, which are the values of the characters. g. [Lan], vol. I, p. 391 ﬀ. 2], p. ). For the sake of conciseness we use the modern language of characters: By deﬁnition, a Dirichlet character mod m is a homomorphism χ : (Z/mZ)× → S 1 , where (Z/mZ)× denotes the group of prime residue classes mod m and S 1 the unit circle in C.