ACTIVITES ADDITIVES ET ANALYTIQUES A LILLE

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ACTIVITES ADDITIVES ET ANALYTIQUES A LILLE — 30 JUIN / 4 JUILLET 2009 Overview of the talks Partitions with relatively prime parts and some congruence by Mohammed El Bachraoui A set A of positive integers is relatively prime if gcd(A) = 1. A partition of n is relatively prime if its parts form a relatively prime set. The number of partitions of n into exactly k parts is denoted by p(n, k) and the number of relatively prime partitions into exactly k parts is denoted by p?(n)(n, k). In our talk we will give explicit formulas for p(n, 3) and prove an identity expressing p?(n)(n, 3) in terms of the Jordan's totient function of order 2. Among other things, using the main theorem of our talk we will obtain the following beautiful result. For any prime p > 3, p?(p)(p, 3) = (p 2 ? 1)/12. Smooth values of binary forms by Antal Balog The following type of problem is raised by some cryptographic application. Let F (X,Y ) be a binary form with integer coefficients, and 1 < P < R be real numbers. There are R2 positive integers X,Y ≤ R such that the prime factors of F (X,Y ) are all < P .

infinite many large

analogue

relatively prime

mobius function

large integer

probleme de waring pour les entiers

arithmetic part

riemann zeta-function

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Publié par

profil-urra-2012

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English

´ ` ACTIVITES ADDITIVES ET ANALYTIQUES A LILLE — 30 JUIN / 4 JUILLET 2009

Overview of the talks

Partitions with relatively prime parts and some congruence byMohammed El Bachraoui A setAof positive integers is relatively prime if gcd(A) = 1. A partition ofnis relatively prime if its parts form a relatively prime set. The number of partitions ofninto exactlykparts is denoted by p(n, k) and the number of relatively prime partitions into exactlykparts is denoted bypΨ(n)(n, k). In our talk we will give explicit formulas forp(n,3) and prove an identity expressingpΨ(n)(n,3) in terms of the Jordan’s totient function of order 2. Among other things, using the main theorem of our talk we will obtain the following beautiful result. For any primep >3, 2 pΨ(p)(p,3) = (p−1)/12.

Smooth values of binary forms byAntal Balog The following type of problem is raised by some cryptographic application. LetF(X, Y) be a binary 2 form with integer coeﬃcients, and 1< R< P be real numbers. There areRpositive integers X, Y≤Rsuch that the prime factors ofF(X, Y) are all< Pdiscuss recent results in this direction.. We

On a sextenary cubic form byValentin Blomer LetN(P) be the number of solutions to x1y2y3+x2y1y3+x3y1y2= 0, y1y2y36= 0,(x1, x2, x3, y1, y2, y3) = 1 6 inside the box [−P, P] . This deﬁnes a singular cubic inP5with an interesting geometry; for example, it contains inﬁnitely many rational planes. We prove an asymptotic formula 3 3−δ N(P) =P Q(logP) +O(P) for some polynomialQof degree 4 with explicit leading coeﬃcient, thereby conﬁrming the Manin con-jecture. The method is a blend of rather diverse techniques and can be generalized in various ways. This is joint work with J. Bruedern.

LineargrowthforChˆateletsurfaces byTim Browning The Manin conjecture predicts the asymptotic growth rate of rational points on rational surfaces. In this talk I will indicate how an upper bound of the expected order of magnitude can be achieved for Chˆateletsurfaces.

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