Functiones et Approximatio Commentarii Mathematici

A remark on the Möbius function

Jerzy Kaczorowski

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Abstract

It is proved that for every positive $B$ there exist real numbers $0=a_0<a_1<\ldots <a_N=1$ and $ \max_{1\leq j\leq N} (a_{j-1}/a_j)\leq\theta<1$ such that $$\limsup_{x\to\infty} \frac{1}{\sqrt{x}}\sum_{j=1}^N\sum_{\theta a_j x<n\leq a_j x} \mu(n) \geq B$$ and $$\liminf_{x\to\infty}\frac{1}{\sqrt{x}}\sum_{j=1}^N\sum_{\theta a_j x<n\leq a_j x} \mu(n)\leq -B,$$ where $\mu(n)$ denotes the Möbius function.

Article information

Source
Funct. Approx. Comment. Math., Volume 39, Number 1 (2008), 61-70.

Dates
First available in Project Euclid: 19 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229696554

Digital Object Identifier
doi:10.7169/facm/1229696554

Mathematical Reviews number (MathSciNet)
MR2490088

Zentralblatt MATH identifier
1228.11149

Subjects
Primary: 11N25: Distribution of integers with specified multiplicative constraints
Secondary: 11N37: Asymptotic results on arithmetic functions 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses

Keywords
Möbius function Mertens conjecture omega estimates

Citation

Kaczorowski, Jerzy. A remark on the Möbius function. Funct. Approx. Comment. Math. 39 (2008), no. 1, 61--70. doi:10.7169/facm/1229696554. https://projecteuclid.org/euclid.facm/1229696554


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