Estimates for the sequence of primes
type: Item
instance of: scholarly article
title: Estimates for the sequence of primes
zbMATH Open document ID: 0688.10004
author: Ulrich Felgner
published in: Elemente der Mathematik
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/141497
review text: Let \(p_ n\) be the nth prime and let \(\pi\) (x) be the number of primes p such that \(p\leq x\). We prove by elementary methods that \[ 0.91 n\cdot \log n<p_ n<1.7 n\cdot \log n,\quad for\quad n\geq 3, \] \[ 0.788 x<\pi (x)\cdot \log x<1.5 x,\quad for\quad x\geq 5, \] and \[ 0.6 x/\log (2x)<\pi (2x)-\pi (x),\quad for\quad x\geq 6. \] These estimates are considerably sharper than those previously obtained by related methods. The basic idea is to work with certain multinomial coefficients instead of the binomial coefficient \(\binom{2n}{n}\) as it is usually done. This also enables us to prove that \[ 2.1^ n<\prod_{p\geq n}p<3^ n,\quad for\quad n\geq 41, \] which yields a one-line proof of Bertrand's postulate.
reviewed by: Ulrich Felgner
Mathematics Subject Classification ID: 11A41
zbMATH DE Number: 4127291
zbMATH Keywords: estimates for n-th prime
zbMATH Keywords: multinomial coefficients
zbMATH Keywords: Bertrand's postulate
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4127291