Analytic Number Theory

Exponential sums over primes in short intervals

A linear equation on k unknowns is called a (k, h)-equation if it is of the form ∑k i=1 aixi = 0, with ai ∈ {−h, . . . , h} and ∑ ai = 0. For a (k, h)-equation E, let rE(n) denote the size of the largest subset of the first n integers with no solution of E (besides certain trivial solutions). Several special cases of this general problem, such as Sidon’s equation and sets without threeterm arithmetic progressions, are some of the most well studied problems in additive number theory. Ruzsa was the first to address the general problem of the influence of certain properties of equations on rE(n). His results suggest, but do not imply, that for every fixed k, all but an O(1/h) fraction of the (k, h)-equations E are such that rE(n) > n1−o(1). In this paper we address the generalized problem of estimating the size of the largest subset of the first n integers with no solution of a set S of (k, h)-equations (again, besides certain trivial solutions). We denote this quantity by rS(n). Our main result is that all but an O(1/h) fraction of the sets of (k, h)-equations S of size k − b √ 2kc + 1, are such that rS(n) > n1−o(1). We also give several additional results relating properties of sets of equations and rS(n).

The purpose of this book is to describe the classical problems in additive number theory, and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools to attack these problems. This book is intended for students who want to learn additive number theory, not for experts who already know it. The prerequisites for this book are undergraduate courses in number theory and real analysis.

Exact order of subsets of asymptotic bases in additive number theory

Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA consisting of all sums of h elements of A. By contrast, in an inverse problem, one starts with a sumset hA, and attempts to describe the structure of the underlying set A. In recent years there has been ramrkable progress in the study of inverse problems for finite sets of integers. In particular, there are important and beautiful inverse theorems due to Freiman, Kneser, Plunnecke, Vosper, and others. This volume includes their results, and culminates with an elegant proof by Ruzsa of the deep theorem of Freiman that a finite set of integers with a small sumset must be a large subset of an n-dimensional arithmetic progression.

Journal Article ERROR TERMS IN ADDITIVE PRIME NUMBER THEORY Get access H. L. MONTGOMERY, H. L. MONTGOMERY Trinity CollegeCambridge Search for other works by this author on: Oxford Academic Google Scholar R. C. VAUGHAN R. C. VAUGHAN The UniversitySheffield Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 24, Issue 1, 1973, Pages 207–216, https://doi.org/10.1093/qmath/24.1.207 Published: 01 January 1973 Article history Received: 30 June 1972 Published: 01 January 1973

of expressing n as the product n = n 1 n 2 .np. Estimates of the error term in the asymptotic formula for the summatory function of d(a; n) are given applying the theory of estimations of multiple exponential sums.

The determination of the minimal s such that all large natural numbers n admit a representation asis an interesting problem in the additive theory of numbers and has a considerable literature, For historical comments the reader is referred to the author's paper [2] where the best currently known result is proved. The purpose here is a further improvement.

One of the most studied problems in additive number theory, Goldbach’s conjecture, states that every even integer greater than or equal to 4 can be expressed as a sum of two primes. In this paper checking of this conjecture up to 4 ⋅ 10 11 4 \cdot {10^{11}} by the IBM 3083 mainframe with vector processor is reported.

An edge‐coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge‐coloured graph on vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of for this question. Very recently, Janzer–Sudakov and Kim–Lee–Liu–Tran independently removed the term in Tomon's bound, showing a bound of . We prove an upper bound of for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the term, and so, it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non‐abelian groups.

An improvement and generalization of Bellman-Shapiro’s theorem on a problem in additive number theory

Lehmer if there was anything known about the representation of an integer A in the form h - In, i-X where all the prime factors of each A,-are of a given form.A search of the literature seemed to indicate that various theorems had been conjectured but none actually proved.fFor example, L. Euler stated without proof that every integer of the form 47+2 is a sum of two primes each of the form 47 + 1.Even the weaker statement that every integer of the form 4/+2 is a sum of two integers which have all their prime factors of the form 4/+1 has not yet been proved.In view of the absence of any definite results in the literature it seems worthwhile to point out that some very interesting theorems can be obtained in an elementary way.This is done in Part I of this paper and the results are summarized in Theorems 1, 2, and 3 below.In Part II we use the method of Viggo BrunJ to prove a general theorem and from this we deduce Theorems 4 and 5 below.

Some Applications of a Theorem of M. Kneser.

In this paper, we consider a question of sum-keeping about a multiplicative subsemigroup and its generator subsets in a semiring, and develop some elementary (collapse) process of the sum-keeping retraction through subsets until one minimal generators subset. As an application, we study and analyze several classical problems in additive number theory on the semiring of non-negative integers by this algebraic and combinatory idea, and provide new proofs in more simple and direct way for several classical results in number theory. Some further questions are also presented and discussed.

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza tion of Dirichlet's L-functions with a generalization of class field the ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory.