Trigonometric polynomials in prime number theory

Rearrangements of Trigonometric Series and Trigonometric Polynomials

Abstract. The paper is related to the following question of P. L. Ul’yanov: is it true that for any 2π-periodic continuous function f there is a uniformly convergent rearrangement of its trigonometric Fourier series? In particular, we give an affirmative answer if the absolute values of Fourier coefficients of f decrease. Also, we study a problem how to choose m terms of a trigonometric polynom...

Some Topics in Prime Number Theory

(ii) p has no divisor other than 1 and itself We denote by P the set of all primes and we write the elements of P in the increasing order as p 1 < p 2 < p 3 < · · · Now consider = p 1 · · · p n + 1. Observe that is an integer > 1. Therefore there exists q ∈ P with q |. Then q = p i for 1 ≤ i ≤ n since q || − 1. Thus q > p n. We started with primes p 1 , · · · , p n and we found a prime q > p n....

Three Topics in Additive Prime Number Theory

We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectu...

New results in comparative prime number theory

Approximation By Trigonometric Polynomials

These notes are prepared as lecture notes exclusively for the participants of this conference only. Any reproduction in any media, or any use for any other purpose of any part of this manuscript, without an expressed written consent of the author is unlawful.thorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conc...