Primes in special intervals and additive problems with such numbers
type: Item
instance of: scholarly article
title: Primes in special intervals and additive problems with such numbers
zbMATH Open document ID: 1108.11064
DOI: 10.1023/A:1023270112701
author: M. E. Changa
published in: Mathematical Notes
publication date: 2004-11-15T00:00:00Z
review text: For integers \(l, r\) with \(1 \leq l \leq r\) and a non-integer real \(a > 1\), let \(\mathcal P_{r, a}\) denote the set of primes \(p\) such that \[ (l - 1) + rm \leq p^a < l + rm \quad \text{for some } m \in \mathbb Z. \] Set \(\delta = 10^{-7}\) and \(\gamma = 6 \cdot 10^{-11}\), and suppose that \(x \geq x_0(a)\), \(r \leq x^{\delta}\), and \(q \leq x^{\delta}\). The main result of the paper under review is the asymptotic formula \[ \sum_{p \in \mathcal P_{r, a}, \, p \leq x} \chi(p)\exp( 2\pi if(p)) = \frac 1r \sum_{p \leq x} \chi(p)\exp( 2\pi if(p)) + O( x^{1 - \gamma/a^2}(\log x)^3), \tag \(*\) \] where \(f(X) \in \mathbb R[X]\) is a polynomial of degree \(\leq 4\) and \(\chi(n)\) is a Dirichlet character modulo \(q\). The \(O\)-implied constant in (\(*\)) and the value of \(x_0(a)\) depend only on the distance from \(a\) to the nearest integer. The author uses (\(*\)) to derive an asymptotic formula for the counting function of \(\mathcal P_{r, a}\) and for the numbers of solutions of several additive problems with unknowns restricted to primes from \(\mathcal P_{r, a}\). For example, he obtains an asymptotic formula for the number of representations of a large odd integer as the sum of three primes from \(\mathcal P_{r, a}\), with \(r, a\) as above.
reviewed by: Angel V. Kumchev
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11L20
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 2115578
zbMATH Keywords: distribution of primes
zbMATH Keywords: exponential sums over primes
zbMATH Keywords: additive problems with primes
MaRDI profile type: Publication
full work available at URL: a:1023270112701
OpenAlex ID: W2937173
description: scientific article; zbMATH DE number 2115578