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Mathematics
Lai Duc Thinh
On the Number of Divisors in an Angle
(Presented by Academician I. M. Vinogradov on 29 XII 1962)
In (⁸) results were given concerning the sum
\[ \sum_{\substack{N(\alpha)\le x\\ \varphi_1\le \arg \alpha\le \varphi_2}} \tau(\alpha), \]
where \(\alpha\) is an integer of an imaginary quadratic field, and \(\tau(\alpha)\) is the number of divisors of the number \(\alpha\), with a remainder term equal to \(O\left(x^{2/3+\varepsilon}\right)\).
In the present paper we shall consider the same problem in real quadratic fields. Let \(\alpha\) be an integer of a real quadratic field \(k\) with discriminant \(d\); \(\tau(\alpha)\) is the number of divisors of \(\alpha\). As in (⁸), we note that
\[ \sum_{|N(\alpha)|=n} \tau(\alpha)=O\left(n^\varepsilon\right) \]
for sufficiently large \(n\). Put
\[ \omega(\alpha)=\frac{1}{2\ln \eta}\ln\left|\frac{\alpha}{\alpha'}\right|, \]
where \(\eta>1\) is the fundamental unit of the field, and \(\alpha'\) is conjugate to \(\alpha\). From each set of all numbers associated with \(\alpha\) we choose \(\alpha\) so that \(0\le \omega(\alpha)<1\). In this note results are given concerning the sum
\[ \sum_{\substack{|N(\alpha)|\le x\\ \omega(\alpha)\le v<1}} \tau(\alpha). \tag{1} \]
We shall consider two functions:
\[ Z(s)=\sum_{\alpha}\frac{1}{|N(\alpha)|^s},\qquad Z(s,q)=\sum_{\alpha}\frac{\lambda^q(\alpha)}{|N(\alpha)|^s}, \]
where \(\alpha\) runs through all integral non-associated numbers of the field \(k\), \(s=\sigma+it\), \(\sigma>1\), and \(\lambda(\alpha)=e^{2\pi i\omega(\alpha)}\), \(q\ne 0\) is a rational integer. For the remainder term of the sum (1), here too we shall obtain \(O\left(x^{2/3+\varepsilon}\right)\). Next we shall generalize the problem to \(\tau_\nu(\alpha)\), namely, we shall compute the sums
\[ \sum_{\substack{|N(\alpha)|\le x\\ \omega(\alpha)\le v<1}} \tau_\nu(\alpha), \tag{2} \]
where \(\tau_\nu(\alpha)\) is the number of solutions of the equations \(\alpha_1,\alpha_2,\ldots,\alpha_\nu=\alpha\), when \(\alpha_1,\alpha_2,\ldots,\alpha_\nu\) independently run through all non-associated divisors of \(\alpha\), so that \(\tau_2(\alpha)=\tau(\alpha)\). It should be noted that in imaginary fields we can also consider a sum analogous to the sum (2), and obtain the same estimates as here for the sum (2).
§ 1.
\[ \sum_{|N(\alpha)|\le x}\tau(\alpha). \]
With the aid of Theorem 101 from (¹) it is easily proved that the fundamental ideal of every quadratic field is the principal ideal \((\sqrt d)\). On this basis, as a special case of Theorem 154 from (³), we obtain the following theorem:
Theorem 1. The function \(Z(s)\), defined for \(\sigma>1\), can be analytically continued to the entire \(s\)-plane, satisfies the equation
\[ \left(\frac{\sqrt d}{\pi}\right)^s \Gamma^2\left(\frac{s}{2}\right) Z(s) = \left(\frac{\sqrt d}{\pi}\right)^{1-s} \Gamma^2\left(\frac{1-s}{2}\right) Z(1-s) \tag{3} \]
and has a pole of the first order at the point \(s=1\) with residue \(2\ln\eta/\sqrt d\).
Theorem 2. In the strip \(-\varepsilon\leqslant \sigma \leqslant 1+\varepsilon\), for \(|t|\geqslant 1\), we have
\[ Z(s)=O\left(|t|^{1-\sigma+\varepsilon}\right). \tag{4} \]
The proof of the theorem is based on the Phragmén—Lindelöf theorem \((^5)\), on Theorem 1, and on estimates for the function \(\Gamma(s)\).
Theorem 3.
\[ \sum_{|N(\alpha)|\leqslant x} \tau(\alpha) = \frac{4\ln^2\eta}{d}\,x\ln x + \left(\frac{4c\ln\eta}{\sqrt d}-\frac{4\ln^2\eta}{d}\right)x + O\left(x^{3/5+\varepsilon}\right), \tag{5} \]
where
\[ c=\lim_{s\to 1}\left\{Z(s)-\frac{2\ln\eta}{(s-1)\sqrt d}\right\}. \]
Proof. Applying the lemma on partial sums of a Dirichlet series \((^4),\) Ch. III) under the assumption \(x=[x]+\frac12\), which entails no loss of generality, we obtain
\[ \sum_{|N(\alpha)|\leqslant x} \tau(\alpha) = \frac{1}{2\pi i} \int_{1+\varepsilon-iT}^{1+\varepsilon+iT} Z^2(s)\frac{x^s}{s}\,dx + O\left(\frac{x^{1+\varepsilon}}{T}\right). \]
Moving the path of integration to the contour
\(L\{1+\varepsilon-iT,\,-\varepsilon-iT,\,-\varepsilon+iT,\,1+\varepsilon+iT\}\), taking into account (3), (4), and Lemma 4, item 1, Ch. IV from \((^4)\), and computing the residue
\(\operatorname*{Res}_{s=1} Z^2(s)\frac{x^s}{s}\), we obtain the required result by putting
\[ T=x^{2/5}. \]
§ 2. \(\displaystyle \sum_{|N(\alpha)|\leqslant x}\tau(\alpha)\lambda_q(\alpha)\).
Theorem 4. The function \(Z(s,q)\), defined for \(\sigma>1\), can be analytically continued to the entire \(s\)-plane and satisfies the equation
\[ \left(\frac{\sqrt d}{\pi}\right)^s \Gamma\left(\frac{s}{2}+\frac{\pi i q}{2\ln\eta}\right) \Gamma\left(\frac{s}{2}-\frac{\pi i q}{2\ln\eta}\right) Z(s,q) = \]
\[ = \left(\frac{\sqrt d}{\pi}\right)^{1-s} \Gamma\left(\frac{1-s}{2}+\frac{\pi i q}{2\ln\eta}\right) \Gamma\left(\frac{1-s}{2}-\frac{\pi i q}{2\ln\eta}\right) Z(1-s,q). \tag{6} \]
The theorem follows directly from equation (44) of \((^2)\).
Theorem 5. In the strip \(-\varepsilon\leqslant \sigma \leqslant 1+\varepsilon\), for \(|t|\geqslant 1\), we have
\[ Z(s,q)=O\left\{(|t|+|q|)^{1-\sigma+\varepsilon}\right\}. \tag{7} \]
The proof is obtained on the basis of Theorem 4, estimates for the \(\Gamma\)-functions, and Theorem 2 from \((^6)\).
Theorem 6.
\[ \sum_{|N(\alpha)|\leqslant x}\tau(\alpha)\lambda^q(\alpha) = O\left\{(x^{1/3}+|q|)^{2+\varepsilon}\right\}. \tag{8} \]
The theorem is proved in the same way as Theorem 3, except that here the function \(Z(s,q)\) is analytic everywhere.
§ 3. The main theorem.
Theorem 7. In real quadratic fields we have
\[ \sum_{\substack{|N(\alpha)|\leqslant x\\ \omega(\alpha)\leqslant v}} \tau(\alpha) = v\left\{ \frac{4\ln^2\eta}{d}\,x\ln x + \left(\frac{4c\ln\eta}{\sqrt d}-\frac{4\ln^2\eta}{d}\right)x \right\} + O\left(x^{2/3+\varepsilon}\right), \tag{9} \]
where \(0<v<1\).
For the proof we shall use the following lemma, which is a modification of a lemma of I. M. Vinogradov from \((^7)\).
Lemma. Let \(v\) be a real number, \(0<2\Delta\leqslant v\leqslant 1-2\Delta\); then there exist two periodic functions \(\overline f(\omega)\) and \(\underline f(\omega)\) with period \(1\), having the following properties:
\[ \overline f(\omega)=1,\quad \text{if } 0\leqslant \omega\leqslant v, \qquad \underline f(\omega)=1,\quad \text{if } \Delta\leqslant \omega\leqslant v-\Delta, \]
\[ 0\leqslant \overline f(\omega)\leqslant 1,\quad \text{if } v\leqslant \omega\leqslant v+\Delta,\quad 1-\Delta\leqslant \omega\leqslant 1, \]
\[ 0\leqslant \underline f(\omega)\leqslant 1,\quad \text{if } v-\Delta\leqslant \omega\leqslant v,\quad 1\leqslant \omega\leqslant 1+\Delta, \]
\[ \overline f(\omega)=0,\quad \text{if } v+\Delta\leqslant \omega\leqslant 1-\Delta; \qquad \underline f(\omega)=0,\quad \text{if } v\leqslant \omega\leqslant 1; \]
\(\overline f(\omega)\) and \(\underline f(\omega)\) are expanded into Fourier series of the form
\[ \overline f(\omega)=\sum_{q=-\infty}^{\infty}\overline a_q e^{2\pi iq\omega}, \qquad \underline f(\omega)=\sum_{q=-\infty}^{\infty}\underline a_q e^{2\pi iq\omega}, \]
where
\[ \overline a_0=v+\Delta,\qquad \underline a_0=v-\Delta, \]
\[ \overline a_q\ll \frac1{|q|},\qquad \underline a_q\ll \frac1{|q|}, \]
\[ \overline a_q\ll \frac1{\Delta^r |q|^{r+1}},\qquad \underline a_q\ll \frac1{\Delta^r |q|^{r+1}}, \]
\[ q\ne 0,\quad r\ne 0\ \text{natural}. \]
With the help of this lemma we have
\[ \sum_{|N(\alpha)|\leqslant x}\tau(\alpha)\,\underline f[\omega(\alpha)] \leqslant \sum_{\substack{|N(\alpha)|\leqslant x\\ \omega(\alpha)\leqslant v}} \tau(\alpha) \leqslant \sum_{|N(\alpha)|\leqslant x}\tau(\alpha)\,\overline f[\omega(\alpha)]. \]
Taking \(\Delta=x^{-1/3}\), \(r=3\), and summing the left- and right-hand sides of these inequalities with respect to \(q\), we shall find formula (9).
§ 4. Generalization for \(\tau_\nu(\alpha)\).
Theorem 8. In quadratic real fields we have
\[ \sum_{\substack{|N(\alpha)|\leqslant x\\ \omega(\alpha)\leqslant v}} \tau_\nu(\alpha) = vxP_{\nu-1}(\ln x)+O\left(x^{\nu/(\nu+1)+\varepsilon}\right), \tag{10} \]
where \(P_\nu(y)\) is a polynomial of degree \(\nu\) in \(y\).
The proof of formula (10) does not differ from the proof of formula (9). For the proof it is first necessary to compose analogues of formulas (5) and (8) for \(\tau_\nu(\alpha)\), namely
\[ \sum_{|N(\alpha)|\leqslant x}\tau_\nu(\alpha) = xP_{\nu-1}(\ln x) + O\left(x^{(2\nu-1)/(2\nu+1)+\varepsilon}\right), \]
\[ \sum_{|N(\alpha)|\leqslant x} \tau_\nu(\alpha)\lambda^q(\alpha) = O\left\{\left(x^{1/(\nu+1)}+|q|\right)^{\nu+\varepsilon}\right\}. \]
Next we apply the lemma with the condition \(\Delta=x^{-1/(\nu+1)}\) and \(r=\nu+1\).
Moscow State University
named after M. V. Lomonosov
Received
27 XII 1962
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