UDC 511.64

MATHEMATICS

A. V. SOKOLOVSKII

THE DISTANCE BETWEEN “NEIGHBORING” PRIME IDEALS

(Presented by Academician I. M. Vinogradov on 26 IV 1966)

Analyzing the classical proof \((^1)\) of the theorem on the difference between “neighboring” prime numbers, it is easy to see that it is based on the knowledge:

a) of the absence of zeros of \(\zeta(\sigma+it)\) in the region
\(\sigma \geqslant 1-A/\ln^a(|t|+2)\) \((a<1)\),

b) of an estimate of \(N(\sigma,T)\)—the number of zeros
\(\rho=\beta+i\gamma\) of the function \(\zeta(\sigma+it)\) in the region
\(\beta \leqslant \sigma;\; 0\leqslant \gamma \leqslant T\).

With the aid of I. M. Vinogradov’s method for estimating trigonometric sums \((^2)\), we prove here Theorems 1 and 2.

Theorem 1. The Dedekind zeta-function \(\zeta_K(\sigma+it)\) of a field of algebraic numbers \(K\) of degree \(n\) has no zeros in the region
\[ \sigma \geqslant 1-A_1/\ln^{2/3}(|t|+2), \]
where \(A_1>0\) depends only on the field \(K\) (cf. \((^{3,4})\)).

Theorem 2. \(\zeta_K(1/2+it)\ll |t|^{\,n/4-c/n^2\ln n};\) \(c\) is an absolute constant.

From Theorem 1, by means of a simple generalization of Hoheisel’s method \((^1)\), we obtain the following theorem.

Theorem 3. Let, as usual, \(\pi(x)\) denote the number of prime ideals of the field \(K\) with norm not exceeding \(x\). Suppose further that
\[ N_K(\sigma,T)\ll T^{b(1-\sigma)}\ln^{c_1}T. \]
Then, for \(\theta>1-1/b\),
\[ \pi(x+x^\theta)-\pi(x)\sim x^\theta/\ln x. \]

Corollary. Since from the estimate
\[ \zeta_K(1/2+it)\ll |t|^{c_0}\ln^{c_2}|t| \]
it follows that
\[ N_K(\sigma,T)\ll T^{2(1+2c_0)(1-\sigma)}\ln^{c_3}T, \tag{5} \]
Theorems 2 and 3 give
\[ \theta>\frac{1+4c_0}{2+4c_0} = 1-\frac{1}{\,n+2-4c/n^2\ln n\,}. \]

We outline the proof of Theorems 1 and 2. In the paper \((^4)\) it is shown that the question of shifting the zeros of \(\zeta_K(\sigma+it)\) reduces to estimating the sum
\[ S= \sum_{\substack{a<a_i<a'\\ (a_1,\ldots,a_n)\in K_1^X\setminus K_0^X}} e^{2\pi i F(a_1,\ldots,a_n)} . \tag{1} \]

Here
\[ F(a_1,\ldots,a_n) = -\frac{t}{2\pi} \ln \prod_{j=1}^{n} \bigl(a_1\alpha_1^{(j)}+\cdots+a_n\alpha_n^{(j)}\bigr); \]
\(a_i\) are rational integers; \(\alpha_1,\ldots,\alpha_n\) is a basis of some integral ideal of the field \(K\); \(\alpha_i^{(j)}\) are the conjugates of \(\alpha_i\); \(K_1^X\) (respectively \(K_0^X\)) is the set of tuples \((a_1,\ldots,a_n)\) such that \(|a_r|\leqslant 2X\) (respectively \(X\)) \((r=1,\ldots,n)\), and the image in \(R^n\) of the number of the field \(K\)
\[ a=a_1\alpha_1+\cdots+a_n\alpha_n \]
belongs to the fundamental region of the field (for more detail see \((^4)\)).

Lemma 1. For any integer \(m\geqslant 1\) and any tuple
\[ (a_1,\ldots,a_n)\in K_1^X\setminus K_0^X \]
\[ \left|\partial^m F(a_1,\ldots,a_n)/\partial a_i^m\right| \leqslant c_4^m(m-1)!\,tX^{-m}. \tag{2} \]

For any fixed \(a_r\) \((r\ne i)\) satisfying \((a_1,\ldots,a_n)\in K_1^X\setminus K_0^X\) and any integer \(m_1\ge 1\), the interval of variation of \(a_i\) can be divided into \(\le c_5^{m_1}\) intervals, for each of which there exists an integer \(m\) \((m_1\le m\le m_1+n)\) such that the inequality

\[ \left|\partial^m F(a_1,\ldots,a_n)/\partial a_i^m\right| \ge c_6^m(m-1)!tX^{-m} \tag{3} \]

holds for all points of this interval.

Proof. It is easily computed that

\[ \frac{\partial^{m_1}F(a_1,\ldots,a_n)}{\partial a_i^{m_1}} = (-1)^{m_1}(m_1-1)!t \sum_{j=1}^{n} \left[ \frac{\alpha_i^{(j)}}{a_1\alpha_1^{(j)}+\cdots+a_n\alpha_n^{(j)}} \right]^{m_1} = \]

\[ = \frac{(-1)^{m_1}(m_1-1)!t} {\left|a_1\alpha_1^{(d)}+\cdots+a_n\alpha_n^{(d)}\right|^{m_1}} \sum_{j=1}^{n} \left[ \frac{\alpha_i^{(j)}}{\left|\alpha_i^{(d)}\right|} \frac{\left|a_1\alpha_1^{(d)}+\cdots+a_n\alpha_n^{(d)}\right|} {a_1\alpha_1^{(j)}+\cdots+a_n\alpha_n^{(j)}} \right]^{m_1}, \]

where

\[ \left| \frac{\alpha_i^{(d)}}{a_1\alpha_1^{(d)}+\cdots+a_n\alpha_n^{(d)}} \right| = \max_{1\le l\le n} \left| \frac{\alpha_i^{(l)}}{a_1\alpha_1^{(l)}+\cdots+a_n\alpha_n^{(l)}} \right|. \]

With the help of Turán’s theorem 2 \((^6)\) and lemma 1 \((^4)\), we obtain the existence of \(m\) \((m_1\le m\le m_1+n)\) such that

\[ \left| \frac{\partial^mF(a_1,\ldots,a_n)}{\partial a_i^m} \right| \ge \frac{c_7^m}{m^n}(m-1)!tX^{-m} \]

for any fixed set \((a_1,\ldots,a_n)\in K_1^X\setminus K_0^X\). Hence the assertions of the lemma follow immediately \((^4)\).

Basic lemma. Let \(m=[\ln t/\ln X]+1\) and \(e^{\ln^{2/3}t}<X<Bt^{(n+1)/n}\). Then

\[ \left| \sum_{\substack{a\le a_i\le a'\\ (a_1,\ldots,a_n)\in K_1^X\setminus K_0^X}} e^{2\pi iF(a_1,\ldots,a_n)} \right| < CX^{1-\gamma/m^2}, \]

where \(C\) and \(\gamma\) depend only on the field \(K\).

The proof almost literally repeats the arguments of I. M. Vinogradov \((^2)\). We note the main specific points.

For \(m>n+4\) put \(Y=[X^{1/3}]\); \(x\) and \(y\) run through the values \(1,2,\ldots,Y\); \(m_0=3m\); \(r=2b\); \(b=lm_0+[m_0(m_0+1)/2+1]\).

Replacing \(a_i\) in (1) by \(a_i+xy\), we obtain, by virtue of inequality (2),

\[ |S|\le \frac{1}{Y^2} \sum_{\substack{a\le a_i\le a'\\ (a_1,\ldots,a_n)\in K_1^X\setminus K_0^X}} |S_{a_i}|+2X^{2/3}, \]

where

\[ S_{a_i}=\sum_x\sum_y e^{2\pi i(A_1xy+\cdots+A_{m_0}x^{m_0}y^{m_0})}, \]

\[ A_s=\frac{1}{2\pi s!}\, \frac{\partial^sF(a_1,\ldots,a_n)}{\partial a_i^s}. \]

As in \((^2)\), the number \(\nu\) of points \((\{A_1\eta_1\},\ldots,\{A_{m_0}\eta_{m_0}\})\) falling into the given small domain of theorem 5 \((^2)\) is estimated by the quantity

\[ (2b c_6^m)^{m_0}X^{m_0(m_0+1)/2} \prod_{[r/2m]+1\le m_r\le 3m-3} X^{m_r-3m+3} \prod_{m\le m_r\le [r/2m]-2} X^{3m-2-3m_r} < \]

\[ < (2b c_6^m)^{m_0}X^{m_0(m_0+1)/2}(1-\delta), \]

where \(m_r\) are those values of \(m\) for which inequality (3) holds; \(\delta>0\) depends only on the field.

Choosing \(k\) and \(l\) sufficiently large in Theorems 5 and 1 \({}^{2}\), we obtain

\[ |S|<CX^{1-\gamma/m^2}. \]

For \(m<n+4\), the estimate for \(|S|\) given in \({}^{4}\) is sufficient. Theorem 1 is now obtained in the usual way \({}^{1,4}\).

With the aid of an estimate for the sum (1) in the interval \(c_7t^{1/3}<X<c_8t^{1/2}\), easily obtained by the method of I. M. Vinogradov \({}^{7,8}\), and of the “approximate functional equation” for \(\zeta_K(\sigma+it)\) \({}^{9}\), Theorem 2 is proved.

Although in special cases (for example, a purely real field \(K\)) the methods of H. Weyl or van der Corput are easily applied to estimating \(\zeta_K(1/2+it)\) (which gives, in Theorems 2 and 3, \(c=1/12\) and \(\delta>(3n+2)/(3x+5)\)), it is not clear how these methods can be used in the case of an arbitrary field.

I express my gratitude to A. A. Karatsuba for valuable advice.

Tashkent State University
named after V. I. Lenin

Received
25 IV 1966

CITED LITERATURE

\({}^{1}\) K. Prachar, Primzahlverteilung, Berlin, 1957.
\({}^{2}\) I. M. Vinogradov, Izv. AN SSSR, Ser. Mat., 29, No. 3 (1965).
\({}^{3}\) I. Yu. Kubilyus, Litovsk. Mat. Sborn., 5, No. 3 (1965).
\({}^{4}\) A. V. Sokolovskii, Izv. AN UzSSR, No. 1 (1966).
\({}^{5}\) A. V. Sokolovskii, Izv. AN UzSSR, No. 3 (1966).
\({}^{6}\) P. Turan, Acta Math. Acad. Sci. Hung., 11, No. 3–4 (1960).
\({}^{7}\) I. M. Vinogradov, Selected Works, Publishing House of the Academy of Sciences of the USSR, 1952.
\({}^{8}\) I. P. Kubilyus, Mat. Sborn., 31 (73), No. 3 (1952).
\({}^{9}\) K. Chandrasekharan, R. Narasimhan, Math. Ann., 152, No. 1 (1963).