Oscillation theorems for primes in arithmetic progressions and for sifting functions

Introduction Differential and integral forms of isoperimetric inequalities Proof of Theorem 1.1 A relation between the distribution of a function and its derivative A variational problem The discrete version of Theorem 5.1 Proof of propositions 1.3 and 1.5 A special case of Theorem 1.2 The uniform distribution on the sphere Existence of optimal Orlicz spaces Proof of Theorem 1.9 (the case of the sphere) Proof of Theorem 1.9 (the Gaussian case) The isoperimetric problem on the real line Isoperimetry and Sobolev-type inequalities on the real line Extensions of Sobolev-type inequalities to product measures on $\mathbf{R}^{n}$ References.

Introduction Statement of results Proof of Theorem 2.1 Proof of Theorem 2.3 Asymptotic analysis of $K_n$ and $R_n$ Proof of Proposition 2.4 Proof of Lemma 4.3 Random initial points Proof of Theorem 2.6: the general case Appendix A. Appendix Bibliography

In this paper, we investigate the existence of fixed points that are not necessarily unique in the setting of extended b-metric space. We state some examples to illustrate our results.

We describe the formalization using the HOL Light theorem prover of Dirichlet's theorem on primes in arithmetic progression. The proof turned out to be more straightforward than expected, but this depended on a careful choice of an informal proof to use as a starting-point. The goal of this paper iis twofold. First we describe a simple and efficient proof of the theorem informally, which iis otherwise difficult to find in one self-contained place at an elementary level. We also describe its, largely routine, HOL Light formalization, a task that took only a few days.

Here we shall use the same notations of the original paper. The references to equations, lemmas, propositions, theorems are with respect to the paper unless stated otherwise. The argument provided for the proof of Proposition 3.1 is not complete. It overlooks the fact that the solution 2-form η(x, t) of the problem (3.1) may contain non-trivial (2,0) and (0,2) parts. Hence Proposition 3.1 needs a correction. We thank Gilles Carron for pointing out this oversight in the original proof of Proposition 3.1 and his comments. We also thank Luen-Fai Tam for helpful related discussions. We refer interested reader to [5] for an alternate proof of Theorem 1.2. Since the preservation of the positivity for solution 2-forms solving the problem (3.1) in the paper, without knowing that the 2-form being (1,1)-type, seems elusive (even for the (1,1) part projection of the 2-form solutions) we replace (3.1) with the following initial boundary value problem on (1,1)form η. Let Ω be a smooth bounded domain in M . For a given point p ∈ ∂Ω , let ν be the unit outward normal in a small neighborhood near p. We

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.

In the previous chapter the proof of Theorem 8.4 was reduced to that of Proposition 14. 2′ which is a multivariate version of Proposition 6.2, and whose proof is based on similar ideas. Here I present those notions and inductive hypotheses whose study leads to the proof of Proposition 14. 2′.

Elementary Methods in Number Theory begins with first course in number for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary number theory. In the second and third parts of the book, deep results in number theory are proved using only elementary methods. Part II is about multiplicative number theory, and includes two of the most famous results in mathematics: the Erdos-Selberg elementary proof of the prime number theorem, and Dirichlets theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive number theory: Warings problems for polynomials, Liouvilles method to determine the number of representations of an integer as the sum of an even number of squares, and the asymptotics of partition functions. Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets.

In this paper we use the Riemann zeta distribution to give a new proof of the Erdös-Kac Central Limit Theorem. That is, if ζ ( s ) = ∑ n ≥ 1 1 n s \zeta (s)=\sum _{n\ge 1} \frac {1}{n^s} , s > 1 , s>1, then we consider the random variable X s X_s with P ( X s = n ) = 1 ζ ( s ) n s , P(X_s=n)=\frac {1}{\zeta (s)n^s}, n ≥ 1. n\ge 1. In an earlier paper, the first author and Adrien Peltzer derived the analog of the Erdös-Kac Central Limit Theorem (CLT) for the number of distinct prime factors, ω ( X s ) , \omega (X_s), of X s , X_s, as s ↘ 1. s\searrow 1. In this paper we show, by means of a Tauberian Theorem, how to obtain the Central Limit Theorem of Erdös-Kac for the uniform distribution from the result for the random variable X s X_s . We also apply the technique to the number of distinct prime divisors of X s X_s that lie in an arithmetic sequence and a local CLT of the type proved by Dixit and Murty [Hardy-Ramanujan J. 43 (2020), 17–23] as well a version of the CLT for irreducible divisors of a monic polynomial over a finite field.

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2 k + p, where k is a positive integer and p is an odd prime. Erdős ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2 k +p, and give a quantitative form of Romanoff’s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdős problem is affirmative.

Si q≥2 est un nombre entier, on designe par Sq(n) la somme des chiffres en base q du nombre entier naturel n et par vq(n) sa valuation q-adique. L’objectif de cet article est d’etudier des sommes d’exponentielles de la forme ∑n≤xexp(2iπ(lmSq(n)+km′Sq(n+1)+θn)) afin d’en deduire certaines proprietes statistiques des nombres entiers n pour lesquels Sq(n) et Sq(n+1) appartiennent a des progressions arithmetiques donnees. Ceci permet d’etendre les resultats obtenus par Gelfond en 1968 et ceux obtenus par Mauduit-Sarkozy en 1996.

This is the first part of a two-part paper. Our paper was motivated by two classical papers: A paper of Sir Charles Wheatstone published in 1844 on representing certain powers of an integer as sums of arithmetic progressions and a paper of J. J. Sylvester published in 1882 for determining the number of ways a positive integer can be represented as the sum of a sequence of consecutive integers. There have been many attempts to extend Sylvester Theorem to the number of representations for an integer as the sums of different types of sequences, including sums of certain arithmetic progressions. In this part of the paper, we will make yet one more extension: We will describe a procedure for computing the number of ways a positive integer can be represented as the sums of all possible arithmetic progressions, together with an example to illustrate how this procedure can be carried out. In the process of doing this, we will also give an extension of Wheatstone’s work. In the second part of the paper, we will continue on the problems initiated by Wheatstone by studying certain relationships among the representations for different powers of an integer as sums of arithmetic progressions.

In 1995, A. Wiles [1,2] announced a proof of Fermat's Last Theorem, which is stated as follows: If \(\pi\) is an odd prime and \(x\), \(y\), \(z\) are relatively prime positive integers, then \(z^\pi\not=x^\pi+y^\pi.\) In this note, a simpler proof of this theorem is offered. It is proved that if \(\pi\) is an odd prime and \(x\), \(y\), \(z\) are positive integers satisfying \(z^\pi=x^\pi+y^\pi,\) then \(x\), \(y,\) and \(z\) are each divisible by \(\pi\). The proof offered by Wiles uses cyclic groups. It is the purpose of this note to offer a proof of this result, with techniques within the grasp of Fermat.

A refinement of Cauchy–Schwarz complexity

We introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain significant improvements on two results. The first improvement is the following. Let l be a positive integer and {u 1 ≥ u 2 ≥ · · · } be a decreasing sequence of probabilities satisfying u n · n 1/(l+1)→∞. Let R = R ω be the random sequence obtained by selecting the natural number n with probability u n . Then every set A of natural numbers with positive upper density contains an arithmetic progression a, a+r, a+2r,..., a+lr of length l + 1 with difference r ∈ R ω . The best previous result (by M. Christ and us) was the condition u n ·n 2−l+1 → ∞ with a logarithmic rate. The new bound is better when l ≥ 4. Our second improvement concerns almost everywhere convergence of double ergodic averages. We construct a (random) sequence {r 1 < r 2 < · · · } of positive integers such that r n /n 2−ϵ → ∞ for all ϵ > 0 and, for any measure preserving transformation T of a probability space, the averages $$\frac{1}{N}\sum\limits_{n < N} {{T^n}{F_1}\left( x \right)} {T^{{r_n}}}{F_2}\left( x \right)$$ converge for almost every x. Our best previous result was the growth rate r n /n (1+1/14)−ϵ → ∞ of the sequence {r n }.