On the missing \(\log\) factor
type: Item
instance of: scholarly article
title: On the missing \(\log\) factor
zbMATH Open document ID: 1410.37007
DOI: 10.1007/978-3-319-74908-2_15
author: Olivier Ramaré
publication date: 2019-01-22T00:00:00Z
review text: This is a survey paper, which is concerned with the problem of establishing upper bounds for sums \[ \sum_{n \leq x} \mu(n) g(x,n). \] Here $\mu$ denotes the Möbius function, and $g$ is a regular function. The author covers the history of the problem and many classical and recent results, with an emphasis on the connection to functional analysis which comes from the fact that the Möbius function can be defined as the convolution inverse of the function $\mathbf{1}$. The writing style is pleasant and leisurely, providing easy access to the topic. The ``missing log factor'' in the title refers to Meissel's equations \[ \sum_{n \leq x} \mu(x) [x/n] = 1 \] and \[ \sum_{n \leq x} \mu(n) \{x/n\} = - 1 + x \sum_{n \leq x} \frac{\mu(n)}{n}, \] where $[\cdot]$ and $\{ \cdot\}$ denote the integer resp. fractional part. In the second equation the left-hand side can be trivially estimated by $n$, and the right-hand side by $n \log n$ -- a missing log factor on the left! For the entire collection see [Zbl 1397.37007].
reviewed by: Christoph Aistleitner
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N56
Mathematics Subject Classification ID: 11A41
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 7005157
zbMATH Keywords: Möbius function
zbMATH Keywords: multiplicative functions
MaRDI profile type: Publication
full work available at URL: 978-3-319-74908-2 15
OpenAlex ID: W2807937319
description: scientific article; zbMATH DE number 7005157