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MATHEMATICS
LAI DUC THINH
ON THE NUMBER OF DIVISORS IN AN ANGLE
(Presented by Academician P. S. Aleksandrov on 23 X 1961)
The estimates for the mean number of divisors of numbers less than \(x\) are well known, namely
\[ \sum_{n \le x} \tau(n) = x \ln x + (2E - 1)x + O(x^\rho), \]
where, with the aid of estimates of \(\zeta(s)\) on the line \(\operatorname{Re} s = 1/2\), one obtains directly the value \(1/3\) for \(\rho\), and this value can be improved. In the present note results are given concerning
\[ \sum_{\substack{N(\alpha) \le x\\ \varphi_1 \le \arg \alpha \le \varphi_2}} \tau(\alpha), \]
where \(\alpha\) is a number of an imaginary quadratic field; the results are based on estimates of functions representable by Hecke series in the critical strip. Let us first note that it is not difficult to prove that
\[ \sum_{N(\alpha)=n} \tau(\alpha) = o(n^\varepsilon) \]
for sufficiently large \(n\).
We shall first consider the fields \(k(\sqrt{D})\), where \(D \le -2\), \(D \equiv 2,3 \pmod 4\). In these fields there are two units \((1,-1)\), and the basis is the system \(1,\sqrt{D}\). Consider the function
\[ Z(s)=\sum_{\alpha} \frac{1}{N(\alpha)^s} = \frac{1}{2}\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}{}' \frac{1}{(m^2-Dn^2)^s}, \]
where \(s=\sigma+it\), \(\sigma>1\), and \(\alpha \ne 0\) runs through all integral nonassociated numbers of the field \(k(\sqrt{D})\).
Theorem 1. The function \(Z(s)\), defined for \(\sigma>1\), is analytically continuable to the whole \(s\)-plane, satisfies the equation
\[ (\sqrt{-D})^{s}\pi^{-s}\Gamma(s)Z(s) = (\sqrt{-D})^{1-s}\pi^{s-1}\Gamma(1-s)Z(1-s) \]
and has a pole of the first order at the point \(s=1\) with residue \(\pi/2\sqrt{-D}\).
Proof. We obtain, starting from the function
\[ \theta\left(\frac{z}{\sqrt{-D}}\right) = \sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty} e^{-(m^2-Dn^2)\pi \frac{z}{\sqrt{-D}}} = \frac{1}{z}\, \theta\left(\frac{1}{z\sqrt{-D}}\right), \]
and from the integral
\[ \int_{0}^{\infty} \left[ \theta\left(\frac{z}{\sqrt{-D}}\right)-1 \right]z^{s-1}\,dz, \]
which gives
\[ 2(\sqrt{-D})^{s}\pi^{-s}\Gamma(s)Z(s) = \frac{1}{s(s-1)} + \int_{1}^{\infty} \left[ \theta\left(\frac{z}{\sqrt{-D}}\right)-1 \right] \left(z^{s-1}+z^{-s}\right)\,dz. \]
Theorem 2.
\[ \lim_{s\to 1}\left(Z(s)-\frac{\pi}{2(s-1)\sqrt{-D}}\right)=C_1, \]
\[ C_1= \frac{3\pi+32C_{1,1}}{4\sqrt{-D}} + \frac{D-1}{D}\,\frac{\pi^2}{6} - 16DJ(1), \]
where
\[ C_{1,1}=\int_0^{\varphi_0}2\sin^2\varphi\cos^2\varphi\ln\sin\varphi\,d\varphi +\int_{\varphi_0}^{\pi/2}2\sin^2\varphi\cos^2\varphi\ln\cos\varphi\,d\varphi, \]
\[ \varphi_0=\operatorname{arc\,tg}\frac{1}{\sqrt{-D}}, \]
\[ J(1)=\int_1^\infty\int_1^\infty \frac{xy(\{x\}\{y\}-x\{y\}-y\{x\})}{(y^2-Dx^2)^3}\,dx\,dy \]
(\(\{x\}\) is the fractional part of \(x\)).
The proof is based on Theorem A \((^3)\).
Theorem 3. In the strip \(-\varepsilon\leqslant\sigma\leqslant 1+\varepsilon\), for large \(t\) we have
\[ Z(s)=O\left(|t|^{1-\sigma+\varepsilon}\right). \]
The proof of the theorem is based on Theorem 1, on the representation of the function \(\Gamma(s)\), and on the theorem of Phragmén—Lindelöf \((^5)\).
Theorem 4.
\[ \sum_{N(\alpha)\leqslant x}\tau(\alpha) =-\frac{\pi^2}{4D}x\ln x+ \left(\frac{\pi^2}{4D}+\frac{C_1\pi}{\sqrt{-D}}\right)x +O\left(x^{2/3+\varepsilon}\right). \]
For the proof we use Theorem 3 and the lemma on partial sums of the Dirichlet series \((^4)\).
Now define the function \(Z(s,q)\) as follows:
\[ Z(s,q)=\sum_\alpha \frac{e^{qi\operatorname{arc}\alpha}}{N(\alpha)^s} =\sum_\alpha \frac{\lambda^{2q}(\alpha)}{N(\alpha)^s} =\frac12\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}{}' \frac{\left(\frac{m+n\sqrt D}{m-n\sqrt D}\right)^q}{(m^2-Dn^2)^s}, \]
where \(q\ne0\) is an integer rational number, \(s=\sigma+it\), \(\sigma>1\).
Theorem 5. The function \(Z(s,q)\), defined for \(\sigma>1\), admits analytic continuation to the whole \(s\)-plane and satisfies the equation
\[ \left(\frac{\sqrt{-D}}{\pi}\right)^s \Gamma(s+|q|)Z(s,q) = \left(\frac{\sqrt{-D}}{\pi}\right)^{1-s} \Gamma(1-s+|q|)Z(1-s,q). \]
The theorem is proved, as is Theorem 1, on the basis of the function
\[ \theta_q\left(\frac{z}{\sqrt{-D}}\right) \sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty} (m+n\sqrt D)^{2q}e^{-(m^2-Dn^2)\pi z/\sqrt{-D}} = z^{-2q-1}\theta_q\left(\frac{1}{z\sqrt{-D}}\right). \]
As for the function \(Z(s)\), for \(Z(s,q)\) we have the following two theorems.
Theorem 6. In the strip \(-\varepsilon\leqslant\sigma\leqslant 1+\varepsilon\), for large \(t\) we have
\[ Z(s,q)=O\left[(|t|+|q|)^{1-\sigma+\varepsilon}\right]. \]
Theorem 7.
\[ \sum_{N(\alpha)\leqslant x}\tau(\alpha)\lambda^{2q}(\alpha) = O\left[(x^{1/3}+|q|)^{2+\varepsilon}\right]. \]
Theorem 8.
\[ \sum_{\substack{N(\alpha)\leqslant x\\ \varphi_1\leqslant\operatorname{arc}\alpha\leqslant\varphi_2}} \tau(\alpha) = \frac1\pi(\varphi_2-\varphi_1) \left[ -\frac{\pi^2}{4D}x\ln x+ \left(\frac{\pi^2}{4D}+\frac{C_1\pi}{\sqrt{-D}}\right)x \right] +O\left(x^{2/3+\varepsilon}\right). \]
The theorem is proved in the same way as Theorem 8 from \((^1)\), on the basis of the theorem following from Theorem 7 of \((^1)\), in which we assume that \(\overline{f(\varphi)}\) and \(f(\varphi)\) have period \(\pi\) and the coefficients of their Fourier series satisfy the relations
\[ \bar a_q\ll\frac{1}{\Delta^3|q|^4},\qquad a_q\ll\frac{1}{\Delta^3|q|^4}. \]
Now consider the fields \(k(\sqrt D)\), where \(D \equiv 1 \pmod 4\), \(D<-3\). These fields differ from the preceding ones in that their basis consists of the numbers \(1,\dfrac{1+\sqrt D}{2}\). Therefore our functions \(Z(s)\) and \(Z(s,q)\) have the form:
\[ Z(s)=\sum_\alpha \frac{1}{N(\alpha)^s} =\frac12 \sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}{}' \frac{4^s}{\bigl[(2m+n)^2-Dn^2\bigr]^s}, \]
\[ Z(s,q)=\sum_\alpha \frac{e^{2qi\arg\alpha}}{N(\alpha)^s} =\sum_\alpha \frac{\lambda^{2q}(\alpha)}{N(\alpha)^s} =\frac12 \sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty} \frac{4^s\left(\dfrac{2m+n+n\sqrt D}{2m+n-n\sqrt D}\right)^q} {\bigl[(2m+n)^2-Dn^2\bigr]^s}. \]
We shall obtain theorems analogous to the preceding ones; here we state the following:
Theorem 8a.
\[ \sum_{\substack{N(\alpha)\le x\\ \varphi_1\le \arg \alpha\le \varphi_2}} \tau(\alpha) = \frac1\pi(\varphi_2-\varphi_1) \left[ -\frac{\pi^2}{D}x\ln x+ \left(\frac{\pi^2}{D}+\frac{2C_2\pi}{\sqrt{-D}}\right)x \right] +O\left(x^{2/3+\varepsilon}\right), \]
where
\[ C_2=C_2(D)= \lim_{s\to 1}\left(Z(s)-\frac{\pi}{(s-1)\sqrt{-D}}\right). \]
The field \(k(\sqrt{-3})\), unlike the others, has 6 units. Its basis consists of the numbers \(1,\dfrac{1+\sqrt{-3}}{2}\). Taking this into account and arguing as before, using the results of the case \(D \equiv 1 \pmod 4\), we obtain:
Theorem 8b. In the field \(k(\sqrt{-3})\),
\[ \sum_{\substack{N(\alpha)\le x\\ \varphi_1\le \arg \alpha\le \varphi_2}} \tau(\alpha) = \frac3\pi(\varphi_2-\varphi_1) \left[ \frac{\pi^2}{27}x\ln x+ \left(\frac{2\pi C_3}{3\sqrt3}-\frac{\pi^2}{27}\right)x \right] +O\left(x^{2/3+\varepsilon}\right), \]
where \(C_3=\frac13 C_2(-3)\).
For the Gaussian field \(k(i)\), which has 4 units, we can argue as for the field \(k(\sqrt{-3})\), using the results of the case \(D\equiv 2,3 \pmod 4\). But since the Gaussian field has a single ideal class, we have
\[ Z(s)=\sum_\alpha \frac1{N(\alpha)^s} =\sum_a \frac1{N(a)^s} =\zeta(s)L(s,\chi), \]
where \(a\) runs over all integral ideals of the field,
\[ \zeta(s)=\sum_{n=1}^{\infty}\frac1{n^s}, \qquad L(s,\chi)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(2n-1)^s}. \]
This shows that \(Z(s)=\zeta(s)L(s,\chi)\) has a pole of first order at \(s=1\) with residue \(\pi/4\), and
\[ \lim_{s\to 1}\left(\zeta(s)L(s,\chi)-\frac{\pi}{4(s-1)}\right) = C_4 = \frac{\pi E}{4} + \sum_{n=1}^{\infty}\frac{(-1)^n\ln(2n-1)}{2n-1}, \]
where \(E\) is Euler’s constant.
With the aid of this result and arguments analogous to the preceding ones, we obtain:
Theorem 8c. In the Gaussian field \(k(i)\) we have
\[ \sum_{\substack{N(\alpha)\le x\\ \varphi_1\le \arg \alpha\le \varphi_2}} \tau(\alpha) = \frac2\pi(\varphi_2-\varphi_1) \left[ \frac{\pi^2}{16}x\ln x+ \left(\frac{C_4\pi}{2}-\frac{\pi^2}{16}\right)x \right] +O\left(x^{2/3+\varepsilon}\right). \]
Moscow State University
named after M. V. Lomonosov
Received
20 X 1961
References
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- E. Hecke, Lectures on the Theory of Algebraic Numbers, Moscow–Leningrad, 1940.
- A. E. Ingham, The Distribution of Prime Numbers, Moscow–Leningrad, 1936.
- E. Titchmarsh, The Theory of the Riemann Zeta-Function, Moscow–Leningrad, 1953.
- E. Titchmarsh, The Theory of Functions, Moscow–Leningrad, 1951.