3,114 research outputs found

    The ternary Goldbach problem

    Get PDF
    The ternary Goldbach conjecture, or three-primes problem, states that every odd number nn greater than 55 can be written as the sum of three primes. The conjecture, posed in 1742, remained unsolved until now, in spite of great progress in the twentieth century. In 2013 -- following a line of research pioneered and developed by Hardy, Littlewood and Vinogradov, among others -- the author proved the conjecture. In this, as in many other additive problems, what is at issue is really the proper usage of the limited information we possess on the distribution of prime numbers. The problem serves as a test and whetting-stone for techniques in analysis and number theory -- and also as an incentive to think about the relations between existing techniques with greater clarity. We will go over the main ideas of the proof. The basic approach is based on the circle method, the large sieve and exponential sums. For the purposes of this overview, we will not need to work with explicit constants; however, we will discuss what makes certain strategies and procedures not just effective, but efficient, in the sense of leading to good constants. Still, our focus will be on qualitative improvements.Comment: 29 pages. To be submitted to the Proceedings of the ICM 201

    On Some Dynamical Systems in Finite Fields and Residue Rings

    Full text link
    We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove them. We also show that some of them require further tuning

    Double Character Sums over Subgroups and Intervals

    Full text link
    We estimate double sums Sχ(a,I,G)=∑x∈I∑λ∈Gχ(x+aλ),1≤a<p−1, S_\chi(a, I, G) = \sum_{x \in I} \sum_{\lambda \in G} \chi(x + a\lambda), \qquad 1\le a < p-1, with a multiplicative character χ\chi modulo pp where I={1,…,H}I= \{1,\ldots, H\} and GG is a subgroup of order TT of the multiplicative group of the finite field of pp elements. A nontrivial upper bound on Sχ(a,I,G)S_\chi(a, I, G) can be derived from the Burgess bound if H≥p1/4+εH \ge p^{1/4+\varepsilon} and from some standard elementary arguments if T≥p1/2+εT \ge p^{1/2+\varepsilon}, where ε>0\varepsilon>0 is arbitrary. We obtain a nontrivial estimate in a wider range of parameters HH and TT. We also estimate double sums Tχ(a,G)=∑λ,μ∈Gχ(a+λ+μ),1≤a<p−1, T_\chi(a, G) = \sum_{\lambda, \mu \in G} \chi(a + \lambda + \mu), \qquad 1\le a < p-1, and give an application to primitive roots modulo pp with 33 non-zero binary digits

    Note on the Theory of Correlation Functions

    Full text link
    The purpose of this note is to improve the current theoretical results for the correlation functions of the Mobius sequence {μ(n):n≥1}\{\mu(n): n\geq 1 \} and the Liouville sequence {λ(n):n≥1}\{\lambda(n): n\geq 1 \}.Comment: Sixty Six Pages. Keywords: Autocorrelation function, Correlation function, Multiplicative function, Liouville function, Mobius function, von Mangoldt function, Exponential Su
    • …
    corecore