MATHEMATICS

D. N. LENSKOI

ON AN UPPER ESTIMATE OF SOME NUMBER-THEORETIC FUNCTIONS IN FIELDS OF ALGEBRAIC NUMBERS

(Presented by Academician I. M. Vinogradov on 13 XII 1962)

Consider the following problem. Let (K) be a field of algebraic numbers, (\Sigma) the ring of integers of the field (K),

[
\Sigma^t=\underbrace{\Sigma\times\Sigma\times\cdots\times\Sigma}_{t\ \text{times}},\qquad
F(\xi_1,\xi_2,\ldots,\xi_t)\in \Sigma[\xi_1,\xi_2,\ldots,\xi_t]
]

and let (\mathfrak M) be some system of finite sets (\mathfrak U(x)\subset \Sigma^t), depending on a real parameter (x), whose range of variation is (x\ge x_0). Denote by (\pi_f(F,\mathfrak M,x)) the number of those points of (\mathfrak U(x)) at which the ideal ((F(\xi_1,\xi_2,\ldots,\xi_t))) decomposes into a product of no more than (f) (not necessarily distinct) prime ideals of the field (K). It is required to study the behavior of (\pi_f(F,\mathfrak M,x)).

Earlier, upper estimates for this function were obtained in some special cases (the case where the field (K) is the field of rational numbers and (t=1), see ((^1)); the case where (K) is an arbitrary field of algebraic numbers and (F(\xi)=\xi(\xi+\alpha)), see ((^2))). Below an upper estimate for (\pi_f(F,\mathfrak M,x)) is given under sufficiently general assumptions concerning (F(\xi_1,\xi_2,\ldots,\xi_t)) and (\mathfrak M), and with a specially chosen (f) (lemma), and with its aid the corresponding estimates are proved for two previously unconsidered concrete cases with (t\ge 2) (theorem).

We agree on the following notation (besides that introduced earlier): (\mathfrak A) is an ideal of the field (K); (\mathfrak P) is a prime ideal of the field (K); (N\mathfrak A) is the norm of the ideal (\mathfrak A); (R) is the field of rational numbers; (\xi) is a variable element of the ring (\Sigma); (\bar\xi=(\xi_1,\xi_2,\ldots,\xi_t)); (x,z) are real variables tending to infinity; (\Delta,\varepsilon,w) are positive real constants; (\omega_F(\mathfrak A)) is the number of pairwise distinct solutions of the congruence (F(\xi_1,\xi_2,\ldots,\xi_t)\equiv 0\pmod{\mathfrak A}). We shall call (F(\xi_1,\xi_2,\ldots,\xi_t)) strongly primitive if for all (\mathfrak P)

[
\omega_F(\mathfrak P)<(N\mathfrak P)^t.
]

The congruence (\bar\xi\equiv \bar\xi^{(0)}\pmod{\mathfrak A}) means, by definition, the fulfillment of the congruences (\xi_i\equiv \xi_i^{(0)}\pmod{\mathfrak A}) ((i=1,2,\ldots,t)). The constants in the symbol (O) which we shall use may depend on (F(\xi_1,\xi_2,\ldots,\xi_t)) and (\mathfrak M), but do not depend on the other variables when the latter vary in the prescribed domains. Variables and constants may subsequently be supplied with various indices.

Let (T(\mathfrak M,x,\mathfrak A,\bar\xi^{(0)})) be the number of points in (\mathfrak U(x)) for which the congruence (\bar\xi\equiv \bar\xi^{(0)}\pmod{\mathfrak A}) is valid. We shall say that (\mathfrak M) is regular with parameters (\Delta,\varepsilon) if

[
T(\mathfrak M,x,\mathfrak A,\bar\xi^{(0)})=
\Delta\,\frac{x^t}{(N\mathfrak A)^t}
+
O\left(\frac{x^{t-\varepsilon}}{(N\mathfrak A)^{t-1}}\right),
\qquad
0<\varepsilon\le 1
]

[
(x\to\infty,\quad N\mathfrak A\le x^\varepsilon,\quad \bar\xi^{(0)}\in\Sigma^t).
]

Let (\mathfrak M_i={\mathfrak U_i(x)}), (\mathfrak U_i(x)\subset\Sigma) ((i=1,2,\ldots,t)). The system (\mathfrak M) of sets (\mathfrak U_1(x)\times \mathfrak U_2(x)\times\cdots\times \mathfrak U_t(x)) we call the Cartesian product of the systems (\mathfrak M_1,\mathfrak M_2,\ldots,\mathfrak M_t). If all (\mathfrak M_i) are regular with parameters (\Delta_i,\varepsilon_i), respectively, then (\mathfrak M) is regular with parameters (\Delta=\Delta_1,\Delta_2,\ldots,\Delta_t), (\varepsilon=\min \varepsilon_i).

Lemma. Let (\mathfrak M) be a regular system with parameters (\Delta,\varepsilon); let (F(\xi_1,\xi_2,\ldots,\xi_t)) be a strongly primitive polynomial satisfying the following conditions:

1)
[
\sum_{N\mathfrak P<z}\frac{\omega_F(\mathfrak P)}{(N\mathfrak P)^{t-1}}
=
w\frac{z}{\ln z}
+
O\left(\frac{z}{\ln^2 z}\right)
\qquad (z\to\infty).
]

2) (F(\xi_1,\xi_2,\ldots,\xi_t)) is the product of (f) polynomials from (\Sigma[\xi_1,\xi_2,\ldots,\xi_t]), each of which has the property that in (\mathfrak U(x)) there are only (O(x^{t-1}z)) points at which (N(F(\xi_1,\xi_2,\ldots,\xi_t))\le z).

Then
[
\pi_f(F,\mathfrak M,x)
\le
\Delta\Gamma(w+1)2^w e^{-w}
\prod_{\mathfrak P}
\frac{
1-\dfrac{\omega_F(\mathfrak P)}{(N\mathfrak P)^t}
}{
\left(1-\dfrac{1}{N\mathfrak P}\right)^w
}
\frac{x^t}{\ln^w x}
\left(1+O\left(\frac{1}{\ln\ln x}\right)\right)
]
[
(x\to\infty).
]

The proof of the lemma just formulated uses Selberg’s sieve for fields of algebraic numbers (in a form somewhat different from that used by Rieger ((^3))) and the Tauberian theorem with a remainder term for (L)-series ((^4)).

In what follows we shall throughout consider the case when (K=R) or (K=R(\sqrt{-d})) ((-d) is the discriminant of (K), (d>0)). Let us define, for each of these cases, the notion of a homothetically generated system.

A. (K=R). On the real axis choose a segment (\mathfrak U_0) of length (\Delta), and choose on it a point as the center of a homothety with coefficient (x). The set of rational integers represented by points lying in the image of the segment (\mathfrak U_0) obtained under the indicated homothety will be denoted by (\mathfrak U(x)). The system (\mathfrak M={\mathfrak U(x)}) will be called a homothetically generated system with parameter (\Delta). This system will be regular with parameters (\Delta,\varepsilon=1).

B. (K=R(\sqrt{-d})). We shall, in the usual way, represent the integers of the field (R(\sqrt{-d})) in the plane of a complex variable. Let (\mathfrak U_0) be some domain bounded by a closed Jordan curve, which we assume to be rectifiable, and let (\Delta) be the area of (\mathfrak U_0). Fix in the plane some point as the center of a homothety with coefficient (\sqrt{x}). The set of integers of the field (R(\sqrt{-d})), represented by points lying in the image of the domain (\mathfrak U_0) obtained under the above homothety, will be denoted by (\mathfrak U(x)). The system (\mathfrak M={\mathfrak U(x)}) will be called a homothetically generated system with parameter (\Delta). This system is regular with parameters (2d^{-1/2}\Delta,\ \varepsilon=1/2).

Theorem. Let (K=R) or (K=R(\sqrt{-d})); let (n) be the degree of (K); let (\mathfrak M) be the Cartesian product of (t) homothetically generated systems with parameters (\Delta_1,\Delta_2,\ldots,\Delta_t), respectively; and let (F(\xi_1,\xi_2,\ldots,\xi_t)) be a strongly primitive polynomial:
[
F(\xi_1,\xi_2,\ldots,\xi_t)
=
\sum_{i=1}^{m} F_i^{\,l_i}(\xi_1,\xi_2,\ldots,\xi_t),
]
where (F_i(\xi_1,\xi_2,\ldots,\xi_t)) ((i=1,2,\ldots,m)) are pairwise nonassociated absolutely irreducible polynomials from (\Sigma[\xi_1,\xi_2,\ldots,\xi_t]), and
[
l=\sum_{i=1}^{m} l_i.
]

Then

[
\pi_l(F,\mathfrak M,x)\ll
2^{nm}m!a_K\Delta_1\Delta_2\cdots\Delta_t
\prod_{\mathfrak P}
\frac{1-\dfrac{\omega_F(\mathfrak P)}{(N\mathfrak P)^t}}
{\left(1-\dfrac1{N\mathfrak P}\right)^m}
\frac{x^t}{\ln^m x}
\left(1+O\left(\frac1{\ln\ln x}\right)\right)
]

[
(x\to\infty),
]

where

[
a_K=
\begin{cases}
1, & \text{if } K=R,\
(2d^{-1/2})^t, & \text{if } K=R(\sqrt{-d}).
\end{cases}
]

The proof of the theorem is reduced to verifying conditions 1) and 2) of the lemma. Since (K=R,\ R(\sqrt{-d})), the verification of condition 2) presents no difficulty. As for condition 1), for the polynomials under consideration its verification is based on the asymptotic law of distribution of the prime ideals of the field (K) and on an estimate for the number of solutions of the congruence (F(\xi_1,\xi_2,\ldots,\xi_t)\equiv 0\;(\mathfrak P)), which is obtained from the well-known theorem of Lang and A. Weil ((^5)) and the results of Shimura ((^6)).

Saratov State University
named after N. G. Chernyshevsky

Received
12 XII 1962

CITED LITERATURE

(^1) Wang Yuan (in Chinese), Shusyuye tszinchzhan, 3, No. 3, 416 (1957).
(^2) G. Rieger, Zs. reine u. angew. Math., 208, No. 1–2, 79 (1962).
(^3) G. Reiger, J. reine u. angew. Math., 199, No. 3–4, 208 (1958).
(^4) G. Freud, Acta math. Acad. Sci. Hung., 2, No. 3–4, 299 (1951).
(^5) S. Lang, A. Weil, Am. J. Math., 76, No. 4, 819 (1954).
(^6) G. Shimura, Am. J. Math., 77, No. 1, 134 (1955).