A printable version
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 >> R^2 positive integers X,Y<=R such that the prime factors of F(X,Y) are all < P. We discuss recent results in this direction.
Let N(P) be the number of solutions tox_1y_2y_3 + x_2y_1y_3 + x_3y_1y_2 = 0, y_1y_2y_3 \not= 0, (x_1, x_2, x_3, y_1, y_2, y_3) = 1inside the box [-P, P]^6. This defines a singular cubic in P^5 with an interesting geometry; for example, it contains infinitely many rational planes. We prove an asymptotic formulaN(P) = P^3 Q(log P) + O(P^{3-delta})for some polynomial Q of degree 4 with explicit leading coefficient, thereby confirming the Manin conjecture. The method is a blend of rather diverse techniques and can be generalized in various ways.
This is joint work with J. Bruedern.
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 Châtelet surfaces.
This short talk is an addendum to the lecture of Valentin Blomer. An Euler product identity for a multiple Dirichlet series will be proved in detail. It is an essentail ingredient in the computation of the leading constant in the joint work with Blomer, but also has some independent interest, as there may be many further applications.
An important problem in number theory is to study the distribution of the non-trivial zeros of the Riemann zeta-function which, if one is willing to assume the Riemann Hypothesis, all lie on a vertical line. It is relatively easy to count how many of these zeros lie in a large interval, so the average spacing between consecutive zeros is easy to compute. However, it is a difficult and interesting problem to show that there are many consecutive zeros that have spacings larger or smaller than average.
In this talk we will exhibit the existence of infinite many large gaps between consecutive zeros of the Riemann zeta-function.
Soit F un corps fini à q éléments et soit p sa caractéristique. Soit k un entier >= 2.
Très approximativement, le problème de Waring pour l'anneau F[T] est analogue au problème de Waring pour les entiers. On veut représenter les polynômes M de F[T] comme sommesM = M_{1}^{k}+\ldots+ M_{s}^{k} (1)avec M_{1},..., M_{s} dans F[T], pour un certain entier s > 0. Ce n'est pas toujours possible, ce qui conduit à introduire l'anneau S(F[T],k) formé par les polynômes M de F[T] admettant une représentation (1). On veut alors prouver l'existence d'un entier w = w(q,k) tel que pour s\geq w, tout M\in{\mathcal S}(F[T],k) admette une représentation (1). Si un tel w n'existe pas, on pose w(q,k) = infini. On a là un analogue de ce qui est connu comme le problème de Waring facile sur Z. Pour avoir un analogue du problème de Waring classique, on s'intéresse aux représentations (1) où les polynômes M_{1},..., M_{s} vérifient les conditions de degrék deg M_{i} < k + deg M. (2)De telles représentations sont dites strictes. Soit S^*(F[T],k) l'ensemble des M dans S(F[T],k) admettant une représentation stricte comme somme de puissances k-ièmes. Soit g(q,k), resp. G(q,k), le plus petit entier s, s'il existe, tel que tout P de S(F[T],k), resp., tout P de S(F[T],k) de degré assez grand, est une somme stricte de s puissances k-iè mes. Si un tel s n'existe pas, on pose g(q,k) = infinity, resp. G(q,k)= infinity.
Dans cet exposé, je donnerai des conditions suffisantes pour que les égalités S^*(F[T],k) = S(F[T],k) = F[T] aient lieu ainsi que des bornes pour les nombres w(q,k), g(q,k) et G(q,k). Je parlerai aussi de quelques cas particuliers oû les conditions suffisantes ne sont pas réalisées, à savoir, le cas des exposants k = p^r+1 et le cas de l'exposant 7 en caractéristique 2.
Given an integer B>=2, an integer n >= 2 is said to be B-smooth if all its prime factors are <= B. Let P(n) stand for the largest prime factor of n >= 2, with P(1)=1. For each integer n >= 2, let delta(n) be the distance to the nearest P(n)-smooth number, that is to the nearest integer whose largest prime factor is no larger than that of n, with delta(1)=1. We study the properties of the function delta(n) and in particular the behavior of the sums \sum_{n <= x} delta(n) and \sum_{n <= x} 1/delta(n).
Generalizations are considered.
This is joint work with Nicolas Doyon.
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_Psi(n) (n, k). In our talk we will give explicit formulas for p(n, 3) and prove an identity expressing p_Psi(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_Psi(p) (p, 3) =( p^2 - 1 ) / 12.
We discuss bounds on extremal sets for problems like those below:
1) What is the largest subset of (Z/nZ)^r that does not contain an arithmetic progression of length k?
2) What is the largest subset of [1,...,n]^r that does not contain a solution of x+y=z (i.e. which is a sum free set)?
3) Colour the elements of [1,...,n]^r red and blue. How many monochromatic Schur triples are there?
4) What is the largest dimension so that the Hilbert cube a_0+{0,a_1}+{0,a_2}+ ...+{0,a_d} is a subset of the squares?
Le théorème des nombres premiers dans les progressions arithmétiques nous informe que les nombres premiers se distribuent de façon uniforme dans les progressions arithmétiques mod q. Toutefois, en faisant une étude plus fine, nous observons l'existence d'un certain biais dans cette distribution, il semble y avoir certaines classes d'équivalence mod q qui contiennent plus de premiers. Ce phénomène est appelé biais de Chebyshev et a été étudié en détail par plusieurs auteurs, notamment par Rubinstein et Sarnak qui ont donné un cadre théorique pour étudier ces questions. Nous proposons ici une formule asymptotique pour prédire ce fameux biais et son comportement quand q est assez grand. Nous proposons aussi quelques résultats numériques.
Il s'agit d'un projet conjoint avec Greg Martin.
Let A and B be two rather dense subsets of [1,N]. Let C be the set of the integers of the form ab+1, with a in A and b in B. We prove that C always contains an element which is divisible by a prime greater than N^{2-o(1)}, provided that the natural densities of A and B are very close to 1.
Work in collaboration with C. Stewart.
Let H be an atomic monoid, C a non-negative integer, a in H a non-unit and z, z' two factorizations of a (in other words, two product decompositions of a into irreducible elements of H). A finite sequence (z_0, ... , z_s) of factorizations of a is called a C-chain of factorizations from z to z', if z_0 = z and z' = z_s and for every i in [1,s], z_i arises from z_{i-1} by replacing at most C atoms of z_{i-1} by at most C new atoms. The catenary degree c(H) of H is the smallest possible C >= 0 such that for all a in H each two factorizations of a can be concatenated by a C-chain. By definition, the monoid is factorial if and only if its catenary degree equals zero.
Suppose that H is a Krull monoid with finite class group G (e.g., the ring of integers of an algebraic number field). Then the catenary degree of H equals the catenary degree of the associated monoid of zero-sum sequences over G. This simple lemma allows to study the catenary degree of H with methods from additive group theory and combinatorial number theory. We will provide lower and upper bounds for c(H) in terms of G.
This is joint work with David J. Grynkiewicz and Wolfgang A. Schmid.
I will report on work in progress with Vicky Neale concerning the problem of representing integers as a sum of bracket polynomials. In particular I shall sketch a proof that every sufficiently large integer is a sum of s numbers of the form n[n\sqrt{2}], provided s is big enough.
Bilinear sums crop up inevitably in the study of exponential sums a la Vinogradov. In particular, the study of exponential sums over prime numbers - essential for approaches to the ternary Goldbach conjecture, for example - leads to sums of the kindsum_n sum_m Lambda^{(r)}(n) Lambda^{(s)}(m) e(alpha n m) e^{- nm/x},where Lambda^{(k)}(n) = Lambda(n) . (log n)^{k-1}.
The traditional way of dealing with such sums is via Linnik's dispersion method. This requires the additional use of a small sieve if we wish a truly good bound. After a brief review, we will how to use a different approach that connects this sum to the large sieve, or, rather, to Plancherel's theorem itself. This leads to improvements in certain ranges.
We split the remainder term in the asymptotic formula for the mean of the Euler phi function into two summands called the arithmetic and the analytic part respectively. We show that the arithmetic part can be studied in a pure arithmetic way. In contrast, study of the analytic part heavily depends on the analytic properties of the Riemann zeta function and on the distribution of its non-trivial zeros in particular.
Joint work with K. Wiertelak.
A classical theorem due to Linnik gives a bound for the least prime number in an arithmetic progression. Lagarias, Montgomery and Odlyzko gave a generalization of this result to any number field. Their proof relies on some results about the distribution of the zeros of the Dedekind zeta function (zero free regions, Deuring-Heilbronn phenomenon). In this talk, I will present some new results about these zeros. As a consequence, we are able to prove an effective version of the theorem of Lagarias, Montgomery and Odlyzko.
The talk will be mainly concerned with some conclusions on sums of cubes of almost primes. Amongst others, I will mention a result obtained recently in a joint work with J. Bruedern which states that every sufficiently large integer can be written as the sum of eight cubes of natural numbers having not more than two prime factors.
A theorem of Erdös-Kác shows that, properly normalized, the number of prime factors of an integer is approximately distributed like a standard Gaussian random variable. The proof due to Rényi-Turán may be interpreted as explaining this result by an unusual type of Poisson approximation; this, in turns, leads to a very close analogy with the conjectured structure of moments of the Riemann zeta function, as suggested by Keating-Snaith, with random permutations playing the role of random matrices. For the analogue question for polynomials over a finite field, the analogy can be pushed further and reveals itself as an interesting (though rather special) case of the large-conductor limit in the Katz-Sarnak philosophy.
This is joint work with A. Nikeghbali, Universität Zürich.
Dans cet exposé, on étudie la répartition et la répartition harmonique des fonctions L de puissances symétriques d'une forme primitive. Les deux cas de carrés et de cubes symétriques seront explicitement formulés.
A heuristic for the size of the Möbius inverse is obtained by means of matrix algebra, using the spectral theorem and the concentration of measure phenomenon on high-dimensional spheres. When this heuristic is applied to the summatory function M(x) of the Möbius function mu, it yields a plausibility argument in favor of the Riemann Hypothesis. Some computational evidence bearing on this will be presented. It seems fairly probable that Thomas Jan Stieltjes discovered this approach to Möbius inversion more than a hundred years ago, while working on his claimed proof of RH (which he never published.)
For a given set M of positive integers, Motzkin asked to find the quantity mu(M, which is the supremum of all upper densities \overline\delta(S), where S is a set of nonnegative integers with the property that a in S, b in S implies a-b not in M. This problem is equivalent to two colouring problems in Graph Theory. The first problem is the "Asymptotic T-colouring efficiency" due to Rabinowitz and Proulx and the second problem is the "Fractional chromatic number of the distance graph generated by M" due to Chang, Liu and Zhu. In this talk I will survey on the progress of the density problem and will show the equivalence of the density problem with the colouring problems.
Erdös had many favorite problems on consecutive integers. We will discuss among others the celebrated Erdös-Mirsky conjecture on consecutive values of the divisor function, stating d(x)=d(x+1) for infinitely many integers x and its generalization for arbitrary integer shifts n, that is, whether for a fixed n the equation d(x)=d(x+n) holds for infinitely many values of x. Although (following the first breakthrough by C. Spiro) these problems were solved later by Heath-Brown (for n=1) and G. Pinner (for all values of n), many problems remained open, for example, the analogue of Pinnerīs result when the number of divisors is substituted by the number of different prime divisors (the analogous problem for n=1 was recently solved by J.-C. Puchta and it was generalized for many other values of n by Y. Buttkewitz). Earlier methods did not allow generalizations when the number of divisors or prime divisors is prescribed, due to the parity barrier. It will be presented in the lecture, how (similarly to small gaps between primes) Selbergīs sieve can lead to a unified method to solve these problems.
This is a joint work with S.W. Graham, D. Goldston and C. Yildirim.
A g-Sidon set is a set (typically of integers) such that pairwise sums represent a number at most g times. We are interested in the maximal size of such a set contained in the interval [1, n].
This has a continuous version. Let f be a nonnegative function defined on the interval [0,1]. Assume that the convolution is at most 1 everywhere. How large can int _0^1 f(t) dt be? Call this maximum sigma. Schinzel and Schmidt improved the trivial estimates and conjectured that the extremal function is 1/\sqrt{\pi x}, which would give sigma = 2/\sqrt \pi. This was recently disproved by Máté Matolcsi.
In a joint work with Javier Cilleruelo and Carlos Vinuesa we find an asymptotics for the g-Sidon problem (for large g), expressed in terms of the Schinzel-Schmidt constant sigma.
I present joint work with G. Bhowmik. Let G_r(n) be the number of representations of $n$ as the sum of $r$ primes. Assuming the Riemann Hypothesis we give an explicit formula for sum_{n <= x} G_r(n) and give good upper and lower bounds for the error term. The proof uses the circle method together with a new method to deal with medium arcs.
We present a new technique to deal with zeta regularized products. This techinque applies to the zeta function associated to a class of sequences of complex numbers, called sequences of spectral type. Sequences of spectral type are essentially characterized by requiring the existence of a certain asymptotic expansion for some associated spectral function. The results are of two types. First, for a sequence S of spectral type, we establish some useful relationships among different spectral functions associated to S; we investigate the zeta function zeta(s,S), and we give formulae for poles, residua, special values, and in particular for the coefficients of the Laurent expansion of zeta(s,S) at s=0. We show that they are given by the coefficients of the asymptotic expansion of one of the spectral function associated to S.
Second, we study double sequences. We investigate the decomposability of the zeta function associated to a double sequence with respect to some simple sequence, and we give formulae for the first terms in the Laurent expansion at zero of the zeta function associated to a double sequence. We particularize this technique to the case of sums of sequences of spectral type, and we give two examples concerning some special functions appearing in number theory.
On présente une technique pour étudier les produits regularisés. Cette technique s'applique à la fonction zêta associée à une classe de suites de nombres complexes, dites de type spectral. En particulier, on étude la fonction zêta associée à des suites doubles, et on obtient des formules pour les coefficients du développement en série de Laurent de la fonction zêta prés de zéro.
Let A be a finite subset of the group G = Z^2. For every element b_i of the sumset A + A = {b_0, b_1, ..., b_{|2A|-1}} we denote byD_i = {a-a': a, a' in A; a+a'= b_i} and r_i=|{(a,a'): a+a'= b_i; a, a' in A }|.After an eventual reordering of A+A, we may assume that r_0 >= r_1 >= ...>= r_{|2A|-1}. We defineR_s(A)=|D_0 cup D_1 cup ... cup D_{s-1}| and R_s(k)= max {R_s(A): A subset of G, |A| =k}.In this talk, we examine the case of s = 4 centers of symmetry C = {c_0, c_1, c_2, c_3 }, c_i = b_i / 2. We will show how to obtain the exact value of R_4(k) and we will also describe the structure of extremal sets.
This is joint work with G. A. Freiman.
We study the binary and the ternary Goldbach problems with prime variables from arithmetic progressions with large moduli and establish theorems of Bombieri-Vinogradov's type for the number of solutions. We apply these results to study the Goldbach problems with arithmetic weights attached to some of the variables.
Zeta functions of groups are Dirichlet generating series encoding arithmetic data associated with infinite groups, such as the numbers of (normal) subgroups of finite index, or irreducible complex representations of finite dimension. Similarly, zeta functions of rings encode, for instance, the numbers of subrings (or ideals) of finite index. A prototype of such a generating function is the Dedekind zeta function of a number field, enumerating the numbers of ideals of a finite index in the ring of integers of a number field.
In many interesting cases, these global functions have Euler-type factorizations into local factors, which are known to be rational. In my talk I will present recent work establishing intriguing symmetries satisfied by these rational functions. The proofs of these "local functional equations" bring together deep results from algebraic geometry (viz. Hironaka's resolution of singularities and consequences of the Weil conjectures), techniques from the theory of p-adic integration, and aspects of the combinatorial theory of finite Coxeter groups.
I will present results concerning the local factors of subring zeta functions associated with additively finitely generated rings (and, as a corollary, those of subgroup zeta functions associated with finitely generated nilpotent groups), as well as the representation zeta functions of finitely generated nilpotent groups and certain compact p-adic analytic groups.
Erdös and Renyi claimed and Vu proved that for all h >= 2 and for all epsilon >0, there exists g = g_h(epsilon) and a sequence of integers A such that the number of ordered representations of any number as a sum of h elements of A is bounded by g, and such that |A \cap [1,x]| >> x^{1/h-epsilon}.
In a joint work with Javier Cilleruelo, Sándor Z. Kiss and Imre Z. Ruzsa, we give two new proofs of this result. The first one consists on an explicit construction of such a sequence. The second one is probabilistic and gives g_h(epsilon) << epsilon^{-1}, improving the bound g_h(\epsilon) << \epsilon^{-h} obtained by Vu.
Pdf version Talk When
We discuss a structural approach to subset-sum problems in additive combinatorics. The core of this approach are Freiman-type structural theorems, many of which will be presented through the talk. These results have applications in various areas, such as number theory, combinatorics and mathematical physics. Here are a few recent applications:
(1) If a subset A of {1,2...,n} has more than n^{1/3} log(3n) elements, then some subset of A sums up to a square. (This was conjectured by Erdos in 1986, with partial results by Alon (87), Lipkin (87), Alon-Freiman (88), Sarkozy (93)). This is a result with Hoi Nguyen.
(2) Let p be a large prime. A set B of positive integer is "small" if the sum of the elements of B is less then p. With Hoi Nguyen and Szemeredi, we showed that if a set A of residues mod p is zero-sum free (no subsum equals zero), then A is very close to being "small".
(Part of the talk is based on the speaker recent survey, but we will introduce several new results as well.)
We report on work that provides estimates for the number of rational points on complete intersections having degree large compared to the dimension. For hypersurfaces defined by diagonal equations, the underlying geometry permits sharper estimates to be obtained. Such estimates may be applied in estimating mean values of Weyl sums, in allied problems concerning quasi-diagonal behaviour, and in counting the number of diagonal equations with many solutions.
This is work joint with Per Salberger.
This is joint work with Danilo Bazzanella (Politecnico di Torino) and Alessandro Languasco (Università di Padova).
Let X be large. We will first give a new estimate for the integral moments of primes in short intervals of the type (p, p + h], where p <= X is a prime number and h = o(X).
Then we will use this to prove that for every lambda > 1/2 there exists a positive proportion of primes p <= X such that the interval (p, p + lambda log X] contains at least a prime number, with an explicit bound for the proportion. As a consequence we improve Cheer and Goldston's result (1987) on the size of real numbers lambda > 1 with the property that there is a positive proportion of integers m <= X such that the interval (m, m + lambda log X] contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the proportion of integers m <= X such that the interval (m, m + lambda log X] contains at least a prime number.
We discuss similar results, assuming the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture.
We discuss formulas for special values of the Dedekind zeta functions of real quadratic fields.