A new type of inclusion exclusion principle for sequences and asymptotic formulas for \(\zeta(k)\)
type: Item
instance of: scholarly article
title: A new type of inclusion exclusion principle for sequences and asymptotic formulas for \(\zeta(k)\)
zbMATH Open document ID: 0789.11010
author: Shigeki Akiyama
published in: Journal of Number Theory
publication date: 1994-06-22T00:00:00Z
review text: Let \(\{a(n)\}\) be a sequence of integers defined by a nondegenerate binary linear recurrence. It is shown that there is a constant \(\kappa\) (depending on the recurrence) such that the following asymptotic formula holds for every positive integer \(l\): \[ \log {| a(1^ l)a(2^ l) \cdots a(n^ l) | \over [a(1^ l),a(2^ l),\dots,a(n^ l)]} ={\zeta(l+1) \over 1-\kappa}+O \left( {\omega(n) \over n} \right), \] where square brackets denote the LCM, \(\zeta(s)\) is the Riemann zeta function, \(\omega(n)=\log n\) for \(l=1\), and \(\omega (n)=1\) for \(l\geq 2\). This improves and extends an earlier result by the author [J. Number Theory 36, 328-331 (1990; Zbl 0708.11016)] for the case \(l=1\).
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 484043
zbMATH Keywords: sequence of integers
zbMATH Keywords: binary linear recurrence
zbMATH Keywords: asymptotic formula
zbMATH Keywords: Riemann zeta function
reviewed by: Tom M. Apostol
MaRDI profile type: Publication
full work available at URL: jnth.1993.1072
OpenAlex ID: W1986417433
DOI: 10.1006/JNTH.1993.1072
description: scientific article; zbMATH DE number 484043