Some Sums involving Farey fractions(Analytic Number Theory)

(1)

Some Sums

involving

Farey

fractions

S. Kanemitsu

(Univ.

of

Kinki, Kyushu)

金光滋 (近畿大九州工)

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

Introduction.

For

any

$x\geqq 1$,let

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

sequence

of all irreducible

fractions withdenominators $\leqq x$,arranged in increasing order of magnitude:

$F_{x}= \{\rho_{\mathcal{V}}=\frac{b_{v}}{c_{v}}|0<b_{v^{\leqq}}c\leqq X,$$(vbc_{v})=1\}\gamma$_’

(sequence) of order $x$

.

It isconvenientto supplement $p_{0}= \frac{0}{1}=\frac{b_{0}}{C_{0}}$to $F_{x}$ toform $F_{x}’$ because it is then

easy

to

construct $F_{x+1}’$ from $F_{x}’$ by just inserting all mediants $\frac{b_{v}+b_{v+1}}{C_{v}+C_{v+1}}$ ofsuccessiveterms $\frac{b_{v}}{c_{v}},$$\frac{b_{v+1}}{c_{v+1}}$ in $F_{x}’$

between them

as

long

as

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

.

E. $\mathrm{g}$

.

from

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

we

form $F_{3}^{\mathrm{t}}= \mathrm{t}\frac{0}{1},\frac{1}{3},\frac{1}{2}$,$\frac{2}{3’}\frac{1}{1}\}$

.

The number oftermsin theFareyseriesof order $x$ is

$\# F_{x}=\Phi(_{X)\sum_{\chi}X}:=n\leq\varphi()=\frac{3}{\pi^{2}}x^{2}+O(X\log x)$

.

Here, $\varphi(n)$ stands fortheEuler function $\sum_{m\leq n}1$,thenumber of integers

$\leqq n$ that

are

relatively

$(m,n)=1$

primeto $n$,and is equalto thenumber ofterms in $F_{x}\mathrm{w}\mathrm{h}\mathrm{o}\mathrm{S}\mathrm{e}$denominator$=n$

.

By $Q_{n}$

we

denote the setof all pairs ofconsecutivetermsin $F_{x}’$:

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

.

E. $\mathrm{g}$

.

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

.

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

.

In [7]

we

considered the

sums

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

$(m\in \mathrm{N})$

mainly inthespecial

cases

$m=2,3$

.

We notethe identity

(2)

which follows from the basic relation $b_{v+1}c_{v}-bVC_{v+1}=1$

.

Hence, e.g. $S_{1}(X)=. \sum(\rho v+1-pv)=p_{\Phi}(x)\Phi v(x)=0-11-p\mathrm{o}=-0=1$

.

Ingeneral, $S_{m}(n)$

can

be thought of

as

the m-th

power

momentofthedifferencesofconsecutive

Farey fractions. On thesemoments thefollowing theorem holds.

Theorem

1.

For $narrow\infty$, wehave

(i) $S_{2}(n)= \frac{2}{\zeta(2)n^{2}}\{\log n+\gamma+\frac{1}{2}-\frac{\zeta’}{\zeta}(2)\}+\frac{4U(n)\log n}{n^{3}}+o(\frac{\log n}{n^{3}}\mathrm{I}$

(ii) $S_{3}(n)= \frac{2\zeta(2)}{\zeta(3)n^{3}}+\frac{3}{\zeta(2)n^{4}}\{\log n+\gamma-\frac{1}{4}-\frac{\zeta’}{\zeta}(2)+2\sigma 2(2)c^{(}(n)2\}1)+\frac{12U(n)\log n}{n^{5}}+o(\frac{\log n}{n^{5}}\mathrm{I}$ (iii) $S_{4}(n)= \frac{2\zeta(3)}{\zeta(4)n^{4}}+\frac{1}{n^{5}}\{\frac{4\zeta(2)}{\zeta(3)}+3\zeta(3)c(n)3\}\mathrm{t}1)$

$+ \frac{20}{3\sigma(2)n^{6}}\{\log n+\gamma-\frac{13}{30}-\frac{\zeta’}{\zeta}(2)+3\zeta^{2}(2)C((2n)1)3+\sigma(2)\sigma(3)c^{(2)}(n)2\}+\frac{40U(n)\log n}{n^{7}}+o(\frac{\log n}{n^{7}}\mathrm{I}$

and

_for

$m\geqq 5$

(iv) $S_{m}(n)= \frac{2\zeta(m-1)}{\zeta(m)n^{m}}+\frac{m}{\zeta(m-1)nm+1}\{\zeta(m-2)+2\zeta^{2}(m-1)c_{m-1}^{(1)}(n)\}$

$+ \frac{m(m+1)}{n^{m+2}}\{\frac{\zeta(m-3)}{3\zeta(m-2)}+\zeta(m-2)c)(1(m-2n)+\sigma(m-1)c_{m-2}^{(2}()n)\}+\frac{35\theta’\log n}{\zeta(2)n^{m+3}}+o(\frac{1}{n^{m+3}}\mathrm{I}$

where $\gamma$ denotes Euler’sconstant,

$\theta’$ is 1 or$0$accordingas$m=5$or$m>5$,

andfor

${\rm Res}>1$,

$c_{s}^{\langle\prime)}(n)is$

defined

as the absolutelyconvergentseries $c_{s}^{(r)}(n)= \sum_{m=1}^{\infty}\frac{\mu(m)}{m}\overline{B\prime}(\frac{n}{m}\mathrm{I}\cdot$ Finally,

$U(n):= \sum_{m5n}\frac{\mu(m)}{m}\overline{B_{1}}(\frac{n}{m})$,

where $\mu(m)$ denotes the$M\ddot{o}bi\mathcal{U}s$function, $\overline{B_{1}}(x)$ denotes the periodic Bemoulli polynomial

of

degree 1. (Note that$\overline{B_{1}}(\mathrm{o})=-\overline{B_{1}}(1)=-\frac{1}{2}=B$ )

$1$

.

Remark1.Theorem 1 givesveryprecise descriptionofasymptotic behavior of these

sums.

Theorem 1 refines fromerresultsofMikolas, Hall, Lehner-Newman, Kanemitsuet al andprovides

a

direct relationship between the

sums

$S_{m}(n)$ involvingFareyfractions and the

error

term$U(n)$ of the summatory function of$\mathrm{E}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{r}^{\dagger}\mathrm{s}$ function.

BetweenTheorem 1 and the results ofMaierthereis

a

closeconnection

as

follows.We put

(3)

Hence$\delta_{\Phi(n)}=\delta_{0}=0,$$\delta_{1}=\frac{1}{n}-\frac{v}{\Phi(n)}$ etc.

Also followingMaier,

we

put

$s_{\mathit{8}^{h}},= \Phi()-2\sum_{0v=}^{n}(\delta)vg(\delta)^{h}v+1$ ’ $g,$

$h\geqq 0$

.

Noting that $\delta_{v+1^{-}}\delta_{v}=(c_{v}c_{v+1})-1-\Phi(n)^{1}-$, _$v<\Phi(n)$,

we

have $s_{2,0^{-}}S_{1,1}= \frac{1}{2}s(2n)-\Phi(n)^{1}-$,

whence byTheorem 1,(i)itfollows that

$\Phi(n\sum_{v=0}^{)-}1(\delta_{v})^{2}=s_{2},0(=_{S)}0,2=s1,1^{+\frac{1}{\zeta(2)n^{2}}}(\log n+\mathit{7}+\frac{1}{2}-\frac{\zeta^{1}}{\zeta}(2))-\frac{1}{2\sigma(2)n^{2}}+o(\frac{\log n}{n^{3}}\mathrm{I}$

$=s_{1,1}+ \frac{1}{\zeta(2)n^{2}}(\log n+\gamma+\frac{1}{2}-\frac{\zeta^{\mathrm{t}}}{\zeta}(2))+o(\frac{\log n}{n^{3}})$

.

Theestimate of $s_{2,0}$ should be difficult.Infact,it

was

known toFranel that

$s_{2,0}= \sum_{=0}^{-}\Phi(n)v1(\delta)v2=O(n^{-1\epsilon}+)\Leftrightarrow RH$.

However, regarding $s_{2,1}$,Theorem 1 already givesa veryprecise asymtotic formula: $s_{2,1}= \frac{1}{6}S_{3}(n)+O(n-4)=\frac{\zeta(2)}{3\zeta(3)n^{3}}+O(n^{- 4})$.

Corollary. The identity

$\sum_{j=0r}^{m-2}\sum\infty=2\sum_{k=1}^{r}\frac{2(\begin{array}{l}j+m+j1\end{array})}{r^{m+j+}1k^{m-j- 1}}=1$

$(k,r)=1$

holds. Combining this with thereciprocity laws

_of

Sitaramachandrarao and Sivaramasarma [14],

[15], weobtain identities like

(1) $\backslash$

$\sum_{r=1}^{\infty}\gamma^{-3}\sum_{k=1}^{r}k^{-}1\frac{5}{4}=\zeta(4)$,

(2) $\sum_{r=1}^{\infty}\gamma^{-}\sum 4k-2\frac{945\zeta^{2}(3)}{\pi^{6}}k=1r=-\frac{1}{3}$.

Remark

2.

(1)representsthe value at $s=3$ of thezeta-function $H(s)= \sum_{n=1}^{\infty}n\sum_{1}^{n}-Sk^{-1}k=$

consideredbyMatsuoka [12],while(2)representsthevalueat $s=4,$ $z=2$ of the zeta-function

$H(s,z)= \sum_{n=1}^{\infty}n^{-}S\sum_{=k1}^{\hslash}k^{-}z$ considered by Apostol-Vu [1]. Zagier[16]hasproved thatthevalues of

multiple zeta-functions

can

in generalbeexpressed

as

linearcombinations ofproducts of values of

(4)

determination ofthecoefficients.

Theorem

2

(Refinmentof

a

theorem ofMikol\’as[13]). Wehave

_for

$narrow\infty$,

(i) $\mathrm{o}\mathrm{t}\sum_{v--1}^{n)}p^{-}va=\frac{\zeta(a)}{(a+1)\zeta(a+1)}n^{a+1}-\frac{1}{a-1}\Phi(n)-\zeta(a)C)(1(an)n^{a}+Oa(n\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}\{a-1,1\}\theta n)$,

where $\theta=\theta(a)=0,$ $a\neq 2$, $\theta(2)=1$

.

(ii) $\Phi \mathrm{t}\sum_{v=1}^{)}\rho_{v}-1=\Phi(n)\{n\}^{a}\log n+\mathit{7}-\frac{1}{2}-\frac{\zeta’(2)}{\zeta(2)}+O(n\log n)$

.

Theorem3 (Refinmentof

a

theorem of Hand-Dumir[2]). Wehave

_for

$narrow\infty$,

$\sum_{(k,k’)\in Q_{n}}\frac{1}{k^{2}k’}$ $= \frac{1}{\zeta(2)n}\{\log n+\gamma+1-\frac{\zeta’(2)}{\zeta(2)}\}+\frac{4U(n)\log n}{n^{2}}+o(\frac{\log n}{n^{2}})$

.

Theorem3 is

a

special

case

oftheformula for$(k,k^{} \sum_{)\in Q_{n}}\frac{1}{k^{a}k’} (a\in \mathrm{N})$.

..$\cdot$.

Theorem4 (Refinmentof

a

theoremofKanemitsu[7],whichinturnis

a

refinementof that

ofLehner-Newman[10]$)$

.

(i) For $0\leqq a,$ $b$,

$\sum_{(k,k’)\in Q_{n}}ka(k’)^{b}=_{C_{a}n-},\frac{\pi^{2}}{6}b(a+b+2ba++2)CU(a,bn)n+a+b+2o(n(\log n)\theta^{n})a+b+1$,

where $\theta’’=1$,

if

$0<b \leqq\frac{1}{2}$ and $\theta’’=0$

if

_$b=0,$$b> \frac{1}{2}$, and

$c_{a,b}= \frac{6}{\pi^{2}}\{\frac{1}{(1+a)(1+b)}-\frac{\Gamma(1+a)\mathrm{r}(1+b)}{\Gamma(3=a+b)}\}$

.

(ii) $\sum_{(k,k’)\in Qn}\frac{1}{kk’(k+k’)}=\frac{12\log 2}{\pi^{2}n}+\frac{2\log 2}{n^{2}}U(n)+o(\frac{1}{n^{2}})$ ;

$(k,k’) \in\sum_{n}\frac{1}{k+k}Q’=\frac{6}{\pi^{2}}(2\log 2-1)n+(1-2\log 2)U(n)+O(1)$;

$\sum_{(k,k’)\in Q_{n}}\frac{kk’}{k+k}’=\frac{11-12\log 2}{3\pi^{2}}n^{3}+(11-12\log 2)\frac{nE(n)}{6}+O(n^{2})$,

where $E(n)= \Phi(n)-\frac{3}{\pi^{2}}n^{2}$.

Remark

3.

In the

course

ofproof ofTheorem 4, (ii),_{we reprove} $\mathrm{G}\mathrm{u}\mathrm{p}\mathrm{t}\mathrm{a}^{\dagger}\mathrm{s}$ identity

$\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{r^{2}(r+k)}=\frac{3}{4}$

.

This

can

beobatinedalso

as

the

case

$a=1\mathrm{o}(k,r)=\mathrm{f}\mathrm{F}\mathrm{l}$

ormula(19) _{for the zeta-function}

$T^{*}(s,z):= \sum_{\gamma}\infty=1\sum_{k=1}^{r}\frac{1}{r^{s}k^{z}(r+k)}=\frac{1}{\zeta(s+Z+1)}T(S,z)$

$(k,r)=1$

consideredbyApostol-Vu [1] (with $T(s,Z)$ the zeta-function defined by Formula(16)).

(5)

including (1), (2) and Gupta’s identity. Also, his theoremenables

one

to

express

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

in terms of

an

infinite series that contains only rational numbers. For the

purpose

of finding the

valueof infiniteseries,Hata’s method [6] is simpler and has widerrangeof applications.

\S 2. Supplementary lemmas

In this section

we

collect lemmas which together with those lemmas in [7]

can

provide

proofs of the theorems stated in\S 1.

Lemma $1^{\mathrm{t}}$

.

For any $u\in \mathrm{C}$ andany $xarrow\infty$,let

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

.

Then forany

$\ell\in \mathrm{N}$ with

$\ell>{\rm Re} u+1$

we

have

$L_{u}(x)=\{$$\frac{\mathrm{l}}{\log u+1x}x^{u+1}+\sigma(-u)+\mathit{7},$’ $u=-u \neq-11+\sum_{=r1}^{\ell}\frac{(-1)^{r}}{r}B\Gamma(\chi)_{X}u+1-r+o(x^{{\rm Re} u^{-\ell}})$

.

Inthe special

case

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

we can

take $l=u+1$ without

error

$tem$, inwhich

caseitis convenienttowrite the

fomula

in the$fom$

$I_{u}(X)= \frac{1}{u+1}\sum^{u+}\gamma=01(-1)^{r(\begin{array}{ll}u +\mathrm{l} r\end{array})u} \overline{B}\Gamma(X)X-r+\zeta+1(-u)$

.

Forinstance,for$k=0$,Lemma 1’just

says

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

.

Inthe

course

of proof ofLemma 1,

we prove

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

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

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

TheCauchy criterionshowsthatthethe integral

on

the RHS represents

an

analytic function for

${\rm Res}>-l$,

so

thatfor ${\rm Res}>-x,$ $\zeta(S)$ is analytic withtheexcceptionof

a

simple poleat $s=1$

.

Since $\ell\in \mathrm{N}$ is arbitrary, this signifies that $\zeta(s)$ is extended analytically

over

the whole plane with

theexcceptionof

a

simple poleat $s=1$

.

Wecan,however, proceed still furthertogive

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

$P=1$ ofFormula(3) to

use

the Lebesgue

dominated

convergence

theorem

on

the groundsthatthepartial

sums

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

(6)

seriesin the

case

$\ell=2$ ofFormula(3) withoutresortingsuch

a

ratherhigh-brow theorem

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 $\mathrm{f}\mathrm{o}\mathrm{r}\zeta(1-s)$ being absolutely convergent for _{${\rm Res}<0$}, this

gives

an

analyticcontinuationof $\zeta(s)$ into ${\rm Res}<0$ in explicit form.

Proof.

In the

case

$l=2$ _the _{above formula has}_the_form

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

Theimproperintegral $\int_{0}^{1}\overline{B}_{2}(t)t^{-_{S-2}}dt$

can

be calculated when-2_{$<{\rm Res}<-1$},to give

$\int_{0}^{1}\overline{B}_{2}(t)t^{-}-2dt=S\frac{1}{1-s}+\frac{1}{s}-\frac{1}{6}\frac{1}{s+1}$

.

Henceinthe

same

region-2$<{\rm Res}<-1$_,weget therepresentation

$\zeta(s)=-\frac{s(s+1)}{2}\int_{0}^{\infty}\overline{B}2(t)t^{-_{S}}-2dt$.

Substituting the absolutely convergent Fourier series $\frac{1}{\pi^{2}}\sum_{n-1}^{\infty}\frac{\cos 2mt}{n^{2}}-$ for $\overline{B}_{2}(t)$,weget

$\zeta(s)=-\frac{s(s+1)}{2}\frac{1}{\pi^{2}}\sum_{n=1}\int_{0}\infty t^{-s-}\cos\infty 22mtdt$

By the changeofvariables $2mt=u$_{this becomes further}

$=-2(2 \pi)s-1s(s+1)\int \mathrm{o}du^{-}-2\cos u\mathcal{U}\sum_{n}S$$\infty\infty=1$nl-s

Hence by

a

formula forMellintransform

$\int_{0}^{\infty}u^{-s}\cos ud_{\mathcal{U}}=\sin\frac{\pi}{2}s\Gamma(1-s)$

we

conclude that

$\zeta(s)=-2(2\pi)S-1S(_{S+}1)(-\sin\frac{\pi}{2}S\mathrm{I}\mathrm{r}(-1-s)\sigma(1-S)$

$=-2(2 \pi)^{s}-1(S-s-1)\Gamma(-1-s)\sin\frac{\pi}{2}s\sigma(1-S)=-2(2\pi)S-1(-S)\mathrm{r}(-s)\sin\frac{\pi}{2}S\sigma(1-s)$,

whence

we

getthe formulain question. $\square$

We continueto state_{further lemmas. As immediate corollaries of Lemma} $1^{\mathrm{t}}$,

we

have

Lemma

11.

For any$a\in \mathrm{R}$and any$r\in \mathrm{N}$wehave

$L_{a}^{*}(_{X},\gamma):=$$\sum_{n\mathrm{S},(}n,\Gamma)x=1n=a\{$

$\frac{\phi(r)\chi^{a}+1}{ra+1}+\zeta(-a)\sum_{d|\gamma}\mu(d)da$, $a\neq-1$

(7)

$+ \sum_{k=1}\ell\frac{(-1)^{k}}{k}\kappa-k\sum a+1(d)\mu\overline{B}k(\frac{X}{d}b^{k1}-+o(\chi^{a}-\ell\sigma\ell(\Gamma d|r))$,

where $\alpha(r)$ is

de.f

ined

before.

Proof $L_{a}^{*}(X, \gamma)=\sum_{dn\mathrm{g}}\sum_{1(n,\Gamma)}\mu(d)=\sum_{d|\gamma}\mu(d\nu^{a}L(a\frac{X}{d})=\sum_{rd|}\mu(d\nu^{a_{L}}a(\frac{X}{d}\mathrm{I}$

$= \sum_{d|r}\mu(dp^{a}\{$ $\frac{1}{a+1}(\frac{X}{d}\mathrm{I}^{a+1}+\sigma(-a),$ $a \neq-1a=-1^{+\sum_{k=1}}\ell\frac{(-1)^{k}}{k}\overline{B}_{k}(\frac{X}{d})(\frac{x}{d})a+k-1$ $\log\frac{x}{d}+\gamma$, $+o(( \frac{X}{d})a-\ell)$ $=\{$ $\frac{x^{a+1}}{a+1}\sum_{d|r}\frac{\mu(d)}{d}+\zeta(-a)\sum_{d|_{\Gamma}}\mu(d)da$, $a\neq-1$ $+ \sum\frac{(-1)^{k}}{k}X^{a+}-\sum_{r}1k(d)\ell(\mu\overline{B}k\frac{X}{d})d|f^{k}-1$ $( \log_{X+_{\mathit{7})}}\sum_{|_{\Gamma}d}\frac{\mu(d)}{d}-\sum\frac{\mu(d)\log d}{d}d|r’ a=-1$ $k=1$

$+o(x^{a-\ell} \sum d^{\ell)}d|\Gamma’$

$\backslash$

,

whence theresultfollows. $\square$

Corolary. Forany $a\in \mathrm{R}$, with $\mathrm{N}\ni\ell>a+1$

$\sum_{v=1}^{\Phi(x}p^{a}v=)\{$

$\frac{1}{a+1}\Phi(x)+\zeta(-a)s-a(_{X)},$ $a\neq-1$ $\ell$

$\sum_{n\cong x}\varphi(n)(\log n+\alpha(n))+_{f}\Phi(x)$,

$a=-1^{+\sum_{r=}}1 \frac{(-1)^{r}}{r}B_{ra1}S-(_{X})+o(x)$

.

$2|r$

Proof.

Substitutingfor $L_{a}^{*}(n)=L_{a}^{*}(n,n)\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}$lemma 11 in$\sum_{\mathcal{V}=1}^{\Phi(x)}p^{\mathit{0}}\mathcal{V}=\sum_{n\mathrm{s}\kappa}\frac{1}{n^{a}}L_{a}^{*}(n)$,

we

get

$\sum_{\mathcal{V}=1}^{\Phi(x)}\rho^{a}v=\sum\frac{1}{n^{a}}\{n\mathrm{g}$

$\frac{\varphi(n)}{n}\frac{n^{a+1}}{a+1}+\zeta(-a)\sum\mu(d|nd)d^{a}$, $a\neq-1$

$\frac{\varphi(n)}{n}(\log n+\alpha(n)+\gamma)$, $a=-1$

$+ \sum_{n\leq x}\frac{1}{n^{a}}\sum_{\gamma=1}l\frac{(-1)^{r}}{r}na+1-_{\Gamma}\sum_{nd|}\mu(d)\overline{B}_{\Gamma}(\frac{n}{d}\mathrm{I}^{d^{a-}+O}1(n\sum_{5x}n\sigma_{a}-\ell(n)a1$

$=\{$

$\frac{1}{a+1}\sum\varphi(n)+\zeta(-a)\sum\sum\mu(d)(\frac{d}{n}\mathrm{I}n5Xd|na,$ $a\neq-1$ $\ell$

$\sum\varphi(n)(\mathrm{l}n\leq x\mathrm{o}\mathrm{g}n+\alpha(n)+\gamma)$

,

$a=-1^{+\sum_{1}\frac{(-1)^{r}}{r}(\begin{array}{ll} ar -1\end{array})}r=B_{r} \sum_{n5xd}\sum_{|n}\mu(d)(\frac{n}{d})^{a-1}$

(8)

$+o( \sum_{n5x}\sigma_{-}\ell(n))$

$=\{$

$\frac{1}{a+1}\Phi(x)+\zeta(-a)\sum da\mathrm{M}n\mathrm{f}\mathrm{i}x(\frac{X}{n})$,

$\sum\varphi(n)(\log n+\alpha(n))+)\Phi(_{X)}$,

$n\leq x$

whence theresultfollows. $\square$

$a\neq-1$

$\ell$

$a=-1+ \sum_{r=1}\frac{(-1)^{r}}{r}B_{\Gamma}\sum_{d\mathrm{a}}d1-a_{\mathrm{M}}(\frac{X}{d}\mathrm{I}+o(x)$,

$2|r$

Lemma

12.

For the$summatoryfi\mathit{4}nCti_{on}$

of

the

arithmeticalfunction

$t(k):= \sum_{kd|}\frac{\mu(d)}{d}\log\frac{k}{d}$

wehaveas $xarrow\infty$

$\sum_{k\mathrm{s}X}t(k)=\frac{X}{\zeta(2)}\{\log X-1-\frac{\zeta’(2)}{\zeta(2)}\}+H(X)\log x+O(\log_{X})$

.

where

$H(x)= \sum_{k5x}\emptyset(k)-\frac{X}{\zeta(2)}$

.

Thislemmais

an

improvementofLemma8 in [7].

Lemma

13.

For $a>1$,

$r=n \sum_{+1}^{\infty}\frac{1}{r^{a}}\sum_{)(k,r}^{\Gamma}k=1=1\frac{1}{k}=\frac{1}{(a-1)\zeta(2)na-1}\{\log n+\gamma+\frac{1}{a-1}-\frac{\zeta’(2)}{\zeta(2)}\}+\frac{U(n)}{n^{a}}\log n+o(\frac{\log n}{n^{a}}\mathrm{I}\cdot$

Lemma 14. For $a\geqq 2,$ $b\geqq 2$,

$\sum_{r=n+1}^{\infty}\frac{1}{r^{a}}\sum_{|dr}\frac{\mu(d)}{d^{b}}=r\frac{1}{(a-1)\zeta(b+1)na-1}+\frac{c_{b}^{(1)}(n)}{n^{a}}+\frac{(1-\alpha)ac_{b-1}((2)n)}{2n^{a+1}}+o(\frac{(\log n)^{\beta}}{n^{a+2-\alpha}})$ ,

where $\alpha=[\frac{2}{b}],$ $\beta=[\frac{3}{b}]$

.

Lemma15. For $a\geqq 2,$ $b\geqq 2$,

$\sum_{r=n+1}^{\infty}\frac{1}{r^{a}}(k,r\sum\frac{1}{k^{b}}=\frac{\zeta(b)}{(a-1)\zeta(b+1)n^{a}-1}+\frac{1}{n^{a}}\{k=r)1=1\frac{c_{b}^{(1)}(n)}{n^{a}}-\frac{\alpha}{a\zeta(2)}\}+\frac{(1-\alpha)}{2n^{a+1}}\{a\zeta(b)c^{(}-(b1n)2)-\frac{(b-2)\beta}{(\alpha+1)\zeta(2)}\}$

$+o( \frac{(\log n)^{\beta}}{n^{a+2-\alpha}})$.

Lemma

16.

Putting$A_{j}(n):=2r \sum_{=2}m\sum_{k=1}^{r}\frac{1}{r^{m+j+1}k^{m-}j-1}$, we canexpress $S_{m}(n)$ as

$(k,r)=1$

(9)

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