Effective and noneffective results on certain arithmetical functions
type: Item
instance of: scholarly article
title: Effective and noneffective results on certain arithmetical functions
zbMATH Open document ID: 0428.10023
DOI: 10.1016/0022-314X(80)90068-2
author: R. Balasubramanian
author: K. Ramachandra
published in: Journal of Number Theory
publication date: 1980-01-01T00:00:00Z
review text: Let \(a\geq 2\) be an integer, \(l,m\) be complex numbers, \(\alpha=\tfrac12-\tfrac12 a\), \[ F(s)= (\xi(s))^a (\xi(2s))^l (\xi(3s))^m = \sum_{n=1}^\infty a_n n^{-s}, \] \[ M(x) = \sum \text{Re}\,s\{F(s)x^s/s\},\quad N(x) = \sum_{n\leq x} a_n,\quad E(x) = N(x)-M(x). \] It is proved that there exists an effective constant \(X_0 = X_0(a,l,m,\varepsilon)\) such that for \(X\geq X_0\) \[ \int_X^{(a+1+\varepsilon)/2} | E(x)x^{-\alpha}|^2 \,dx/x > X^{-\varepsilon}. \] In particular, this result implies that, if \(\mathcal D(n)\) is the number of square free divisors of \(n\), \[ E(x)= \sum_{n\leq x} \mathcal D(n) - x/\zeta(2)(\log x +2\gamma - 1 - 2\zeta'(2)/\zeta(2)), \] then for \(x\geq x_0\) \[ \max_{X\leq x\leq X^{(3+\varepsilon)/2}} | E(x)/x^{1/4 - \varepsilon}| \geq 1. \]
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3666897
zbMATH Keywords: omega estimates
zbMATH Keywords: number of squarefree divisors
reviewed by: G. A. Kolesnik
MaRDI profile type: Publication
cites work: On the zeros of a class of generalized Dirichlet series. II.
cites work: Q3879321
cites work: Q3937489
cites work: On the zeros of generalised Dirichlet series. V.
cites work: Q4198832
cites work: An Ω-Theorem of the ‘Non-Effective’ Type
cites work: Q5556571
Recommended article: Some results on arithmetical functions
Recommended article: Q3545946
Recommended article: Some effective inequalities between usual arithmetic functions
Recommended article: Two results on arithmetical functions
Recommended article: Q4655310
Recommended article: Q3545955
Recommended article: Q2999436
Recommended article: On non-entire arithmetic functions
description: scientific article; zbMATH DE number 3666897