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October, 2000 Some sums involving Farey fractions II
Shigeru KANEMITSU, Takako KUZUMAKI, Masami YOSHIMOTO
J. Math. Soc. Japan 52(4): 915-947 (October, 2000). DOI: 10.2969/jmsj/05240915

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).

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Shigeru KANEMITSU. Takako KUZUMAKI. Masami YOSHIMOTO. "Some sums involving Farey fractions II." J. Math. Soc. Japan 52 (4) 915 - 947, October, 2000. https://doi.org/10.2969/jmsj/05240915

Information

Published: October, 2000
First available in Project Euclid: 10 June 2008

zbMATH: 1012.11018
MathSciNet: MR1774636
Digital Object Identifier: 10.2969/jmsj/05240915

Subjects:
Primary: 11B57
Secondary: 11N37

Keywords: Euler's function , exponential sums , Farey series

Rights: Copyright © 2000 Mathematical Society of Japan

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Vol.52 • No. 4 • October, 2000
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