Some sums involving Farey fractions. II. (English) Zbl 1012.11018

Kanemitsu, Shigeru; Kuzumaki, Takako; Yoshimoto, Masami

J. Math. Soc. Japan 52, No. 4, 915-947 (2000).

Let \(F_x=\{\rho_\nu\}\) denote the Farey series of order \(\lfloor x\rfloor\) with real \(x\geq 1\) arranged in increasing order of magnitude, and let \(Q_x\) denote the set of pairs \((c_\nu,c_{\nu+1})\) of their denominators. In Part I [ibid. 34, 125-142 (1982; Zbl 0464.10008)] S. Kanemitsu, R. Sitaramachandrarao and A. Sivaramasarma considered the sums \(s_m(x)=\sum_{(c_\nu,c_{\nu+1})\in Q_x} (c_\nu,c_{\nu+1})^{-m}\). In the present paper they improve the asymptotic formulae for \(s_m(x)\) given in the above paper. Then they prove a precise asymptotic for the unsymmetric form of \(s_m(x)\), namely for sums \(\sum_{(c_\nu,c_{\nu+1})\in Q_x} c_\nu^{-a}c_{\nu+1}^{-1}\) thereby improving results by R. J. Hans and V. Chander [Res. Bull. Panjab Univ., New Ser. 15, 353-356 (1965; Zbl 0135.10402)] and finally they improve results of their own and results by M. Mikolás [Acta Univ. Szeged., Acta Sci. Math. 13, 93-117 (1949; Zbl 0035.31402); ibid. 14, 5-21 (1951; Zbl 0042.27014)] on asymptotic behaviour of sums \(\sum_{\rho\in F_x} \rho^{-a}\) for \(a>0\).

Reviewer: Štefan Porubský (Praha)

MSC:

11B57	Farey sequences; the sequences \(1^k, 2^k, \dots\)
11N37	Asymptotic results on arithmetic functions

Keywords:

Farey series; exponential sums; Euler’s function

Citations:

Zbl 0464.10008; Zbl 0035.31402; Zbl 0042.27014; Zbl 0135.10402