数理解析研究所講究録
1091 巻 1999 年 1-81

指標付き約数問題(Divisor Problem with Characters)
名古屋大学多元数理

Let$\chi$

谷川好男 (Yoshio Tanigawa)be a Dirichlet character modand let$k$be the function$r_{a}(n, \chi)$defined by$r_{a}(n, \chi)=\sum_{d|n}\chi(d)d^{a}$

where

$a$

is a fixed real number. When$\chi$

is identically 1, this function is aclassical divisor function usually written by$a=0$ and$\chi$,$\sigma_{a}(n)$. On the other hand, whenis the Kronecker symbol corresponding to$r_{a}(n, \chi)=\frac{1}{4}r(n)$$\mathbb{Q}(i)$, then.

with $r(n)=\#\{(x, y)\in \mathbb{Z}^{2}|x^{2}+y^{2}=n\}$. Hence we can also consider$r_{a}(n, \chi)$as a generalization of the counting function of the lattice points on a circle.
We shall consider the sum of$r_{a}(n, \chi)$and the mean square of its error term.

Before stating our results, we shall recall some known results about thesum of$\sigma_{a}(n)$. Put$\triangle_{0}(x)=\sum_{xn\leq}/(\mathrm{o}n)-x(\log X+2\gamma-1)-1/\sigma 4$

where$\gamma$,

is Euler’s constant and the prime on the summation means that thelast term is to be halved if$x$

is an integer. In 1956, Tong proved that$\int_{1}^{x_{\Delta_{0}}}(x)^{2}dX=\frac{1}{6\pi^{2}}\frac{\zeta^{4}(3/2)}{\zeta(3)}X^{3/2}+O(X\log^{5}X)$.

2and the$O$-term was improved to $o(X\log^{4}x)$ by Preissmann in 1988. For$-1<a<0$ , we define.

$\Delta_{a}(x)=\sum_{xn\leq}/(n)-\zeta(1\sigma a-a)x-\frac{\zeta(1+a)}{1+a}x1+a+\frac{1}{2}\zeta(-a)$

The mean square formula of$\mathrm{K}\mathrm{i}_{\mathrm{t}1(}\cdot \mathrm{h}\mathrm{i}$,and$\mathrm{i}\mathrm{m}\tau$)$\mathrm{r}0\mathrm{V}\wedge \mathrm{e}\mathrm{d}\mathrm{b}\mathrm{v}\vee$$\Delta_{a}(x)\mathrm{f}\mathrm{o}\mathrm{r}-1/2<a<0$

Meurman$l\mathfrak{n}$was first considered byfact: Meurman$\mathrm{t}A\mathrm{r}2\rceil \mathrm{o}\mathrm{r}\mathrm{o}\mathrm{v}\mathrm{e}\sim \mathrm{d}$that$\int_{1}^{X}|\Delta_{a}(X)|2dX$

$=$

$\{$

$\frac{\zeta(3/2-a)\zeta(3/2+a)\zeta 2(3/2)}{(6+4a)\pi^{2}\zeta(3)}X^{3/+}2a+O(X)$

for $-1/2<a<0$$\frac{\zeta^{2}(3/2)}{24\zeta(3)}X\log X+O(X)$

for $a=-1/2$$O(X)$

for $-1<a<-1/2$

For the case-l $<a<-1/2$ , the more precise formula had already beenobtained by S. Chowla in 1932. In [1], he showed that$\int_{1}^{X}|\triangle_{a}(x)|^{2}d_{X}=\frac{\zeta^{2}(1-a)\zeta(-2a)}{12\zeta(2-2a)}X+O(x^{3/+a}2\log X)$

for-l $<a<-1/2$ . Recently, the last formula was obtained$\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\dot{\mathrm{p}}$endently

by Yanagisawa in a somewhat general situation in [3]. He proved that$\int_{1}^{X}\triangle_{a}(\frac{h}{k}X)\Delta_{b}(x)d_{X}$$=$$\frac{1}{2\pi^{2}kh}(\sum_{1n=}^{\infty}\frac{\sigma_{1+}a(hn)\sigma 1+b(kn)}{n^{2}})x$$+O(x^{(3a+b}-\dagger^{-})/2\log X)$.

for-l $<a<0,$ $-1<b<0,$ $a+b<-1$ and $h>0,$ $k>0,$ $(h, k)=1$ .
The aim of this note is to show the corresponding mean square formulafor$r_{a}(n, \chi)$. We assume throughout that$\chi$

is a non-trivial Dirichlet charactermod . In our case, the error term of the sum function of$k$$r_{a}(n, \chi)$by.

$\triangle_{a}(x, \chi)=\sum_{n\leq x}/(n, x)-Lr_{a}(1-a, x)X+\frac{1}{2}L(-a, x)$

is defined

3Theorem 1.

Let $-1<a,$ $b<0$ be real numbers and let $k\ll\sqrt{X}$ .

Suppose that$\chi_{1}$and$\chi_{2}$are primitive Dirichlet characters modwith the$k$same parity.
(i) For $a+b>-1$ , we have$\int_{1}^{X}\Delta_{a}(x, x1)\Delta b(X, x2)dX$$C_{1}X^{(+}3a+b)/2+O( \min(k^{2}X, kx(\log X)^{2}))+O(k^{\max(\frac{6}{4}}+\frac{a+b}{2},1)x)$

$=$

,

where
$C_{1}$$=$$\frac{\tau(\chi_{1})_{\mathcal{T}(x}2)k^{(b}-1+a+)/2}{2\pi^{2}(3+a+b)}$$\cross\frac{\zeta(\frac{3-a-b}{2})L(\frac{3+a-b}{2},\overline{x}1)L(\frac{3-a+b}{2},\overline{\chi}_{2})L(\frac{3+a+b}{2},\overline{x}1\overline{x}2)}{L(3,\overline{\chi}_{1}\overline{\chi}2)}$,

and$\tau(\chi)=\sum_{n=1}^{k-1}\chi(n)e2\pi in/k$.

(ii) For $a+b=-1$ , we have$\int_{1}^{\mathrm{x}_{\triangle_{a}}}(_{X}, \chi_{1})\Delta-1-a(x, \chi 2)dX$$=$$C_{2}X \log X+O(\min(k2x, kX(\log X)2))$if$O( \min(k^{2}x, kX(\log X)2))$otherwise$\{$$\chi_{2}=\overline{\chi}_{1}$with$C_{2}= \frac{\chi_{1}(-1)L(2+a,\overline{x}1)L(1-a,x_{1})}{24\zeta(3)}\prod_{p|k}\frac{p^{2}}{p^{2}+p+1}$

Theorem 2.
$\chi_{1}$and$\chi_{2}$.

Let-l $<a,$ $b<0,$ $a+b\geq-1$ and $k\ll\sqrt{X}$ . Suppose thatare primitive Dirichlet characters mod$k$with the opposite parity.

Then we have$\int_{1}^{X}\triangle_{a}(x, x1)\Delta b(x, x2)d_{X}$$=$$O( \min(k^{2}x, kX(\log X)2))+O(k^{\max(}\frac{5}{4}+\frac{a+b}{2},1)x)$.

4

For the case $a+b<-1$ , we have the following theorem.
Let-l $<a,$ $b<0,$ $a+b<-1$ and $k\ll\sqrt{X}$ . Let

Theorem 3.
$\chi_{2}$and$\chi_{1}$be non-trivial Dirichlet characters mod . Then we have$k$$\int_{1}^{x_{\triangle_{a}(X}},$$\chi 1)\Delta_{b}(X, \chi_{2})d_{X}=C_{3}X+O(k^{2}\log kX^{()/}3+a+b2\log X)$$wi\iota ere$$C_{3}= \frac{L(1-a,\chi 1)L(1-b,\chi 2)L(-a-b,x_{1\chi)}2}{12L(2-a-b,\chi_{1\chi)}2}$

In these theorems, the$\mathrm{O}$.

-constants do not depend on the modulus$k$.

Outline of the proof of Theorems 1 and 2.
For the proof of Theorems 1 and 2, we need the$\triangle_{a}(x, \chi)$. Let$\phi(s)$be the generating function of$\mathrm{v}_{\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{o}}\mathrm{i}$$r_{a}(n, \chi)$formula for. It is easily seenthat $\phi(s)=\zeta(s)L(S-a, \chi)$ . Let$0$if$\chi$

is an even character,$\delta_{\chi}=\{$

1 if
and

$W(\chi)=(-1)^{\delta_{\chi}}\tau(\chi)/\sqrt{k}$$\chi$

is an odd character,. Then the functional equation of$\phi(s)$is givenby$\phi(s)=W(\chi)k^{1}/2+a-s\pi 2S-(1+a)_{\frac{\Gamma((1-s)/2)\mathrm{r}((1+a-s+\delta_{x})/2)}{\mathrm{r}(s/2)\Gamma((s-a+\delta_{x})/2)}\tilde{\phi}}(1+a-s)$

where$\tilde{\phi}(s)=\zeta(s-a)L(_{S},\overline{x})$The$\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\dot{\mathrm{f}\mathrm{i}}$cients of$\tilde{\phi}(s)$.

are given by$\tilde{r}_{a}(n, \chi)=\sum_{|dn}\overline{\chi}(\frac{n}{d})d^{a}$This function plays an important role in.

$\mathrm{v}_{\mathrm{o}\mathrm{r}\mathrm{o}}\mathrm{n}\mathrm{o}\mathrm{i}$formula.

5First we assume$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}-1/2<a<0$. We have$\Delta_{a}(x, \chi)$$=$$-W( \chi)k^{a}/2x\sum(a+1)/2\frac{\tilde{r}_{a}(n,\chi)}{n^{(a+1)}/2}n\infty=1\{\cos\frac{\pi(a-\delta_{x})}{2}Y_{1+a}(4\pi\sqrt{nx/k})$$+ \sin\frac{\pi(a-\delta_{x})}{2}J1+a(4\pi\sqrt{nx/k})+\frac{2}{\pi}\cos\frac{\pi(a-\delta_{x})}{2}K1+a(4\pi\sqrt{nx/k})\}$$=$$\frac{W(\chi)}{\sqrt{2}\pi}k^{a/2+1/4}X^{a/}/4\sum_{=1}2+1\frac{\tilde{r}_{a}(n,\chi)}{n^{3/4+a}/2}n\infty\cos(4\pi\sqrt{nx/k}-(1/4+\delta_{\chi}/2)\pi)$$- \frac{4a^{2}+8a+3}{32\sqrt{2}\pi^{2}}W(\chi)k^{a}/2+3/4/2-X^{a}\sum_{n}^{\infty}1/4=1\frac{\tilde{r}_{a}(n,\chi)}{n^{5/4+a}/2}$$\cross\cos(4\pi\sqrt{nx/k}-(3/4+\delta_{\chi}/2)\pi)+O(k^{a/25/4a//4}+x)2-3$

Here,$\mathrm{Y}_{1+a}(x),$$J_{1+a}(x)$.

and $K_{1+a}(x)$ denote the standard Bessel functions oforder $1+a$ . We also need the sum formula of$\tilde{r}_{a}(n, \chi)$. It is given by$\sum_{n\leq x}/(an, \chi)=\frac{L(1+a,\overline{\chi})}{1+\alpha}\tilde{r}x1+a+\zeta(-a)L(\mathrm{o},\overline{\chi})+\tilde{\Delta}a(n, \chi)$with$\triangle_{a}(x, \chi)\sim$$=$$- \overline{W(\chi)}k^{-}a/2x\sum^{\infty}(a+1)/2\frac{r_{a}(n,\chi)}{n^{()}a+1/2}n=1\{\cos\frac{\pi(a-\delta_{x})}{2}\mathrm{Y}_{1+a}(4\pi\sqrt{nx/k})$$+ \sin\frac{\pi(a-\delta_{x})}{2}J1+a(4\pi\sqrt{nx/k})+\frac{2}{\pi}\cos\frac{\pi(a+\delta_{x})}{2}K1+a(4\pi\sqrt{nx/k})\}$$=$$\overline{\frac{W(\chi)}{\sqrt{2}\pi}}k^{-a/2+1}/_{X}4a/2+1/4\sum_{n=1}^{\infty}\frac{r_{a}(n,\chi)}{n^{3/4+a}/2}\cos(4\pi\sqrt{nx/k}-(1/4+\delta x/2)\pi)$$- \frac{4a^{2}+8a+3}{32\sqrt{2}\pi^{2}}\overline{W(\chi)}k^{-}a/2+3/4_{X}a/2-1/4\sum_{n=1}^{\infty}\frac{r_{a}(n,\chi)}{n^{5/4+a}/2}$$\mathrm{x}\cos(4\pi\sqrt{nx}/k-(3/4+\delta\chi/2)\pi)+O(k-a/2+5/4a/2-X)3/4$

Following Meurman, we get the truncated expression ofabove formulas valid for-l $<a<0$ .

.

$\Delta_{a}(x, \chi)$from the6For-l $<a<0,$ $y\geq 1,$ $X\geq y,$ $Z\geq 2y$ andLemma 1.

, we have$\triangle_{a}(y, x, \chi)+Ra(y, X, z)$$=$$\triangle_{a}(y, \chi)$$y\not\in \mathbb{Z}$$+O(k^{1+}\epsilon y^{-}+/4+a/2k13/4+a/2-y+k4+a/2y-3/4+a/2)1/25/$ ,where$X\mathrm{A}/,$$arrowa(y,$$\chi_{/}^{\backslash }$$=$$W(\chi)_{J^{a/2+}}\sqrt{2}\pi^{n}’ 1/4y^{a/}2+1/4$$\cross\int_{1}^{2}\sum_{n\leq ux}\frac{\tilde{r}_{a}(n,\chi)}{n^{3/4+a}/2}\cos(4\pi\sqrt{ny/k}-(1/4+\delta_{x}/2)\pi)du$and$R_{a}(y, X, Z)$$=$$cW( \chi)\sum_{Zn\leq}ra(n, x)\int 12\int_{u\mathrm{x}}\infty)t-1\mathrm{i}\mathrm{s}\mathrm{n}(\frac{4\pi}{\sqrt{k}}(\sqrt{y}-\sqrt{n}\sqrt{t})dtdu$

with some constant$c$which is independent on .
$k$

From this Lemma , we get Theorems 1 and 2. Note that if$\chi_{1}$and$\chi_{2}$have the opposite parity, there occurs no main term.

Outline of the proof of Theorem 3
formula does not work well. We use the ChowlaIn this case the$\mathrm{v}_{\mathrm{o}\mathrm{r}\mathrm{o}}\mathrm{n}\mathrm{o}\mathrm{i}$Walum’s type formula instead. Namely we have
Lemma 2.
$\Delta_{a}(x, \chi)=-m\sum_{\leq\sqrt{x}}x(m)m^{a}\psi(\frac{x}{m})-x^{a}\sum n^{-a_{P}}(\frac{x}{m})+^{o(}k3/2(\log k)_{X}a/2)n\leq\sqrt{x}$where$\psi(x)$$=x-[x]-1/2$$P(x)$$=$$- \sum_{n\leq x}\chi(n)-\frac{1}{k}\sum_{\leq nk}x(n)n$.

7

The main term comes from the product of the first sums of$\Delta_{b}(x, x_{2})$$\Delta_{a}(x, \chi_{1})$and. The remaining products give the error term. To show this, weneed Yanagisawa’s main lemma.

Lemma 3. Let $f(t)$ and $g(t)$ be piecewise$continuou\mathit{8}$functions of period$A$and of bounded variations. Suppose that$|f(t)|\leq F$,$|g(t)|\leq G$,

and$\int_{0}^{A}f(t)dt=0$.

Then, for any $n\leq\sqrt{X}$ and any sequence of points $\{X_{m}\}$ , and$0<\mathrm{Y}_{m}-X_{m}\ll X_{f}$$\{\mathrm{Y}_{m}\}$with

we have$\sum_{m\leq\sqrt{X}}|\int_{x}^{Y_{m}}f(\frac{x}{m})g(\frac{x}{n})dx|\ll GX\log x(mF(A+\sigma-1(n))+V_{f})$where$V_{f}$

is the total variation of$f$in $[0, A]$ .

The above summation is estimated as $\sigma_{-1}(n)X\log x$ in [3], but it is not

sufficient for our purpose. So we made the dependence on the modulus
explicit in Lemma 3. Note thatand
do not need to be primitive
characters, because we don’t use functional equation in this case.

$k$$\chi_{1}$$\chi_{2}$As in the theorems of Meurman and Chowla-Yanagisawa, the mean square

formula is effectively deduced by$\mathrm{v}_{\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{o}}\mathrm{i}$formula when $a+b\geq-1$ and by

Chowla-Walum formula when $a+b<-1$ . This phenomenon can be expressed
symbolically as8Acknowledgements. I am grateful to Professor Shigeru Kanemitsu andDr. Naoki Yanagisawa for their hearty advices concerning this work.

References[1] S. Chowla, Contributions to the analytic theory of numbers, Math. Z.

35 (1932), 279-299.
[2] T. Meurman, The mean square of the error term in a generalization ofDirichlet’s divisor problem, Acta Arith. 76 (1996), 351-364.
[3] N. Yanagisawa, An asymptotic formula for a certain mean value in adivisor problem, Journal of Number Theory, 73 (1998), 339-358.