Arithmetical problems with numbers of special type
type: Item
instance of: scholarly article
title: Arithmetical problems with numbers of special type
zbMATH Open document ID: 0951.11028
DOI: 10.1007/BF02674886
published in: Mathematical Notes
publication date: 2000-08-02T00:00:00Z
review text: Let \(E_0\) and \(E_1\) denote the sets of positive integers such that the number of ones in their binary representation is even and odd, respectively. The authors state seven theorems, without any proofs, concerning functions like \(\pi_k(X,m,\ell)=\# \{n<X: n\in E_k\) and \(n\equiv \ell\pmod m\}\) and \(\pi_k(X) = \# \{n<X: n\in E_k\}= \pi_k(X,1,0)\), where \(k=0\) or \(k=1\) and \(m, \ell\) are positive integers. We quote here Theorem 1. Suppose that \(m\) is a positive integer such that \(2\leq m \leq \sqrt{X}\). Then the following asymptotic formula holds: \[ \pi_k(X,m,\ell)= \frac 1{2m} X (1+O(X m^{-2})^{-\lambda_4}), \] where \(\lambda_4 = 1-\log 3/\log 4 = 0.207518\dots\) .
reviewed by: László Tóth
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11A63
zbMATH DE Number: 1429641
zbMATH Keywords: binary representation
zbMATH Keywords: asymptotic formula
zbMATH Keywords: Euler's function
author: Anatolii A. Karatsuba
author: B. Novák
MaRDI profile type: Publication
cites work: The distribution of the sum of digits (\bmod𝑝)
full work available at URL: https://doi.org/10.1007/bf02674886
OpenAlex ID: W2015838700
description: scientific article; zbMATH DE number 1429641