Journal of the Mathematical Society of Japan

Some sums involving Farey fractions II

Shigeru KANEMITSU, Takako KUZUMAKI, and Masami YOSHIMOTO

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Abstract

Let Fx denote the Farey series of order [x], i.e. the increasing sequence of irreducible fractions pv(0,1] whose denominators do not exceed x. We shall obtain precise asymptotic formulae for the sum v=1Φ(x)pvZ for complex z and related sums, Φ(x)=Fx coinciding the summatory function of Euler's function. In particular, we shall prove an asymptotic formula for pv-1 with as good an estimate as for the prime number theorem by extracting an intermediate error term occurring in the asymptotic formula for Φ(x).

Article information

Source
J. Math. Soc. Japan, Volume 52, Number 4 (2000), 915-947.

Dates
First available in Project Euclid: 10 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1213107117

Digital Object Identifier
doi:10.2969/jmsj/05240915

Mathematical Reviews number (MathSciNet)
MR1774636

Zentralblatt MATH identifier
1012.11018

Subjects
Primary: 11B57: Farey sequences; the sequences ${1^k, 2^k, \cdots}$
Secondary: 11N37: Asymptotic results on arithmetic functions

Keywords
Farey series Euler's function exponential sums

Citation

KANEMITSU, Shigeru; KUZUMAKI, Takako; YOSHIMOTO, Masami. Some sums involving Farey fractions II. J. Math. Soc. Japan 52 (2000), no. 4, 915--947. doi:10.2969/jmsj/05240915. https://projecteuclid.org/euclid.jmsj/1213107117


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