Some Sums involving Farey fractions

数理解析研究所講究録 958 巻 1996 年 14-22

14

Some Sums involving Farey fractions S. Kanemitsu (Univ. of Kinki, Kyushu) 金光滋

Introduction.

For any

fractions with denominators

$\leqq x$

,

$x\geqq 1$

$(vbc_{v})=1\}\gamma$ ’

$x$

$F_{x+1}’$

from

$F_{x}=F_{[x]}$

denote the sequence of all irreducible

$( \rho_{1}=\frac{1}{[x]},$

$\rho_{\Phi(X)-}1-\frac{1}{[x]}=1)$

$F_{x}’$

$p_{0}= \frac{0}{1}=\frac{b_{0}}{C_{0}}$

to

by just inserting all mediants

between them as long as

$c_{v}+c_{v+1}\leqq \mathrm{X}+1$

. E. . from

$F_{x}$

to form

$\frac{b_{v}+b_{v+1}}{C_{v}+C_{v+1}}$

$F_{x}’$

$x$

because it is then easy to

of successive terms

$F_{2}^{\mathrm{t}}= \{\frac{0}{1},\frac{1}{2},\frac{1}{1}\}$

$\mathrm{g}$

The number of terms in the Farey series of order

we form

stands for the Euler function

$\varphi(n)$

$\sum_{m\leq n}1$

$n$

By

,

, the

and is equal to the number of terms in $Q_{n}$

number of integers

$F_{x}\mathrm{w}\mathrm{h}\mathrm{o}\mathrm{S}\mathrm{e}$

$\mathrm{g}$

$,$

we denote the set of all pairs of consecutive terms in

.

$Q_{4}=\{(1,3),(3,2),(2,3),(3,1)\}$

$(\# Q_{n}=\# FX=\Phi(_{X}))$

$F_{x}’$

.

.

In [7] we considered the sums $(m\in \mathrm{N})$

$S_{m}(x):= \sum_{\chi}(cv’ C)v+1\in Q(Cvc_{v+}1)-m$

mainly in the special cases

$m=2,3$

.

,

in

$\frac{2}{3’}\frac{1}{1}\}$

.

$\leqq n$

denominator

$Q_{x}= \{(c_{v^{c_{v})|\}}},+1\frac{b_{v}}{C_{\mathcal{V}}}1$

,

Finally,

,

denotes the periodic Bemoulli polynomial of

).

Remark 1. Theorem 1 gives very precise description of asymptotic behavior of these sums. Theorem 1 refines fromer results of Mikolas, Hall, Lehner-Newman, Kanemitsu et al and provides

a direct relationship between the sums the summatory function of

$\mathrm{E}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{r}^{\dagger}\mathrm{s}$

$S_{m}(n)$

involving Farey fractions and the error term $U(n)$ of

function.

Between Theorem 1 and the results of Maier there is a close connection as follows. We put $\delta_{v}=p_{\mathcal{V}}-\frac{v}{\Phi(n)}$

,

$v=0,\ldots,\Phi(n)$

16 Hence

$\delta_{\Phi(n)}=\delta_{0}=0,$

$\delta_{1}=\frac{1}{n}-\frac{v}{\Phi(n)}$

etc.

Also following Maier, we put $s_{\mathit{8}^{h}},= \Phi()-2\sum_{0v=}^{n}(\delta)vg(\delta)^{h}v+1$

Noting that

$\delta_{v+1^{-}}\delta_{v}=(c_{v}c_{v+1})-1-\Phi(n)^{1}-$

,

$v{\rm Re} u+1$

$1^{\mathrm{t}}$

.

For any

and any

$u\in \mathrm{C}$

$xarrow\infty$

,

let

$I_{u}(x):= \sum n^{u}n5x$

. Then for any

with

$\ell\in \mathrm{N}$

we have ’

$L_{u}(x)=\{$ $\frac{\mathrm{l}}{\log u+1x}x^{u+1}+\sigma(-u)+\mathit{7},$

In the special case where

$u=-u \neq-11+\sum_{=r1}^{\ell}\frac{(-1)^{r}}{r}B\Gamma(\chi)_{X}u+1-r+o(x^{{\rm Re} u^{-\ell}})$

$u\in \mathrm{N}\cup\{0\}$

,

we can take

$l=u+1$

.

without error $tem$, in which

case it is convenient to write the fomula in the $fom$ $I_{u}(X)= \frac{1}{u+1}\sum^{u+}\gamma=01(-1)^{r(\begin{array}{ll}u +\mathrm{l} r\end{array})u} \overline{B}\Gamma(X)X-r+\zeta+1(-u)$

For instance, for $k=0$ , Lemma 1’ just says

$b(x)= \chi-\overline{B}1(X)-\frac{1}{2}$

.

In the course of proof of Lemma 1, we prove the identity for any

(3)

.

$\ell\in \mathrm{N}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\ell>-{\rm Res}$

$\zeta(s)=A_{-}(Sl)=1+(-1)^{l+1(\begin{array}{l}-S\ell\end{array})-}\int_{1}\infty-S-\ell-\frac{1}{1-s}\overline{B_{\ell}}(t)tdt\sum_{\Gamma=0}\ell(1)\Gamma B_{r}$

$= \frac{1}{s-1}+\frac{1}{2}+\frac{1}{12}s+\cdots+(-1)\ell+1\int_{1}^{\infty}\overline{B}_{f}(t)t^{-s^{-}}\ell dt$

The Cauchy criterion shows that the the integral on the RHS represents an analytic function for ${\rm Res}>-l$

Since

,

so that for

$\ell\in \mathrm{N}$

${\rm Res}>-x,$

$\zeta(S)$

is analytic with the excception of a simple pole at

is arbitrary, this signifies that

the excception of a simple pole at

$s=1$

$\zeta(s)$

$s=1$

.

is extended analytically over the whole plane with

. We can, however, proceed still further to give a

conceptually the simplest proof of the functional equation of the Riemann zeta-fucntion. Actually,

although in a well-known proof one considers the case

$P=1$

of Formula (3) to use the Lebesgue

dominated convergence theorem on the grounds that the partial sums of the Fourier series of

$\overline{B_{1}}(t)$

are bounded, we can prove the following corollary by just employing absolutely convergent Fourier

19 series in the case

of Formula (3) without resorting such a rather high-brow theorem

$\ell=2$

Corollary.

The Riemann zeta-fi4nction satisfies thefunctional equation $\zeta(s)=2(2\pi)s-1\mathrm{r}(1-S)\sin\frac{\pi}{2}S\sigma(1-s)$

The defining Dirichlet series

gives an analytic continuation of

Proof.

In the case

$l=2$

$\zeta(s)$

being absolutely convergent for

$\mathrm{f}\mathrm{o}\mathrm{r}\zeta(1-s)$

into

.

${\rm Res}