Journal of the Mathematical Society of Japan

Some sums involving Farey fractions II

Abstract

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

Article information

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

Dates
First available in Project Euclid: 10 June 2008

https://projecteuclid.org/euclid.jmsj/1213107117

Digital Object Identifier
doi:10.2969/jmsj/05240915

Mathematical Reviews number (MathSciNet)
MR1774636

Zentralblatt MATH identifier
1012.11018

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