Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

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Analytic Number Theory

It is proved that 
lim inf n?8 (p n+1 -p n )<7×10 7 , where p n is the n -th prime. 
Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose

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Bounded gaps between primes

Context The concept of citizen science (CS) is currently referred to by many actors inside and outside science and research. Several descriptions of this purportedly new approach of science are often heard in connection with large datasets and the possibilities of mobilizing crowds outside science to assists with observations and classifications. However, other accounts refer to CS as a way of democratizing science, aiding concerned communities in creating data to influence policy and as a way of promoting political decision processes involving environment and health. Objective In this study we analyse two datasets (N = 1935, N = 633) retrieved from the Web of Science (WoS) with the aim of giving a scientometric description of what the concept of CS entails. We account for its development over time, and what strands of research that has adopted CS and give an assessment of what scientific output has been achieved in CS-related projects. To attain this, scientometric methods have been combined with qualitative approaches to render more precise search terms. Results Results indicate that there are three main focal points of CS. The largest is composed of research on biology, conservation and ecology, and utilizes CS mainly as a methodology of collecting and classifying data. A second strand of research has emerged through geographic information research, where citizens participate in the collection of geographic data. Thirdly, there is a line of research relating to the social sciences and epidemiology, which studies and facilitates public participation in relation to environmental issues and health. In terms of scientific output, the largest body of articles are to be found in biology and conservation research. In absolute numbers, the amount of publications generated by CS is low (N = 1935), but over the past decade a new and very productive line of CS based on digital platforms has emerged for the collection and classification of data.

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What Is Citizen Science?--A Scientometric Meta-Analysis.

In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when sets of different sizes are involed. It is shown that if and pe pδ (e)|A|. Next we exploit the Szemeredi–Trotter theorem in finite fields (also obtained in [8]) to derive several new facts on expanders and extractors. It is shown for instance that the function f(x,y) = x(x+y) from to satisfies |F(A,B)| > pβ for some β = β (α) > α whenever and |A| ~ |B|~ pα, 0 pρ where ρ pδ(e)|A|. This is the finite fields version of a problem considered in [11]. If A is an interval, there is a relation to estimates on incomplete Kloosterman sums. In the Appendix, we obtain an apparently new bound on bilinear Kloosterman sums over relatively short intervals (without the restrictions of Karatsuba's result [14]) which is of relevance to problems involving the divisor function (see [1]) and the distribution (mod p) of certain rational functions on the primes (cf. [12]).

More on the sum-product phenomenon in prime fields and its applications

© Université Bordeaux 1, 1993, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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Integers without large prime factors

Le probleme des diviseurs de Titchmarsh ([19]), consiste en l'evaluation pour / entier ^1, de la somme Γ(,χ; /) definie par T(x\\l)= Σ d(p-l) Kp^x (d(n) designe ici le nombre de diviseurs de rentier «, et p un nombre premier). Linnik fut le premier le resoudre sans utiliser l'hypothese de Riemann §έηέralisee ([15]); la demonstration, fondee sur la methode de dispersion, est tres delicate, mais eile se simplifie beaucoup des qu'on dispose du theoreme de Bombieri-Vinogradov ([2], [20]) qui donne un equivalent en moyenne de la fonction n(x\\q, ) (cardinal de l'ensemble des nombres premiers ^x9 congrus a modulo q). En posant lix E(x; q, a) = n(x\\ q, a) -— et ^ = logx, on montre ([6]):

Sur le problème des diviseurs de Titchmarsh.

par exemple [1, 17-26, 34] on)t permis d'affiner notablement notre connaissancede sa structure. La double motivatio d'un n interet intrinseque manifest et d'une edisponibilite" nouvell dee techniques prometteuses porte done naturellemen a tre"examiner le probleme de la repartition dans les progressions arithmetique des sentiers de S(x, y) (cf. [9 11,, 16, 28, 33] pou ler s principau x travaux anterieursconcernant cette question).Conformement a l'usage, nous designon par W(xs , y) le cardina dl e S(x, y).Nous posons(1.1) W(x,y,a,q):= £ 1.

Entiers sans grand facteur premier en progressions arithmetiques

Exponential sums with monomials

We prove that the 4-rank of class groups of quadratic number fields behaves as predicted in an extension due to Gerth of the Cohen–Lenstra heuristics.

On the 4-rank of class groups of quadratic number fields

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the l-adic Fourier transform introduced by Deligne and studied by Katz and Laumon.

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Étienne Fouvry

Papers

Sur le problème des diviseurs de Titchmarsh.

Entiers sans grand facteur premier en progressions arithmetiques

Exponential sums with monomials

On the 4-rank of class groups of quadratic number fields

Algebraic twists of modular forms and Hecke orbits