MATHEMATICS

E. K. FOGELS

ON THE DISTRIBUTION OF PRIME IDEALS

(Presented by Academician A. I. Mal’cev on 23 V 1961)

Yu. V. Linnik (¹) proved that the least prime number of the arithmetic progression (Dl + l) ([(D,l)=1,\ u=0,1,2,\ldots]) does not exceed (D^c), where (c) (and below (c')) is a suitable absolute positive constant. K. A. Rodosskii (²) gave a simpler proof, and I (³) proved the existence of a prime number of an arithmetic progression in the interval ((x, xD^\varepsilon)) for any positive (\varepsilon \leq c), for all (D \geq D_0(\varepsilon)) and all (x \geq D^{c' \log(c/\varepsilon)}). The purpose of the present note is to report that the following results can be proved by the same method.

Theorem 1. Let (K), (\mathfrak f), (\mathfrak H) denote respectively any algebraic field of degree (n \geq 1), any ideal in the field (K), and any class of ideals modulo (\mathfrak f). Let (D=|\Delta|N\mathfrak f>1), where (\Delta) is the discriminant of the field and (N\mathfrak f) is the norm of the ideal (\mathfrak f). There exists a positive constant (c), depending only on (n), such that for any (x \geq 1) there is, in the interval ((x, xD^c)), a prime number (p=N\mathfrak p), with (\mathfrak p \in \mathfrak H).

For (\mathfrak f=\mathfrak o) (the unit ideal) it follows that any class (in the ordinary sense) of ideals contains a prime ideal with norm (\leq |\Delta|^c).

Theorem 2. There exist absolute constants (c>0), (c'>0), such that for any positive (\varepsilon \leq c), for all discriminants (d) with (|d|=D \geq D_0(\varepsilon)), and for all (x \geq D^{c' \log(c/\varepsilon)}), there is, in the interval ((x, xD^\varepsilon)), a prime number (p) represented by the given primitive (and positive, if (d<0)) binary quadratic form (\psi) of discriminant (d).

The proof of Theorem 1 is based on Lemmas 1–3.

Lemma 1. Let (\zeta(s,\chi)) denote any Hecke function (⁴) with character modulo (\mathfrak f). There exists (c_1) (depending only on (n)) such that in the region

[
\sigma \geq 1-\frac{c_1}{\log D(1+|t|)^{3/4}},
\qquad
(\sigma=\operatorname{Re}s,\ t=\operatorname{Im}s,\ D>D_0>1)
\tag{1}
]

the function (\zeta(s,\chi)) has no zeros with complex (\chi). For at most one real exceptional character (\chi=\chi') in (1) a simple zero (\beta) of the function (\zeta(s,\chi')) is possible; moreover (\beta=1-\delta) is real and

[
\delta > D^{-2n}.
\tag{2}
]

The lemma is proved by the methods of Titchmarsh (⁵) and Page (⁶). Here an important role is played by the estimate

[
F(s)\ll \delta^{-n}D^{\frac12(1-\sigma)}(1+|t|)^{(1+\delta-\sigma)n/2}
]

[
\left(-\delta \leq \sigma \leq 1+\delta,\qquad
0<\delta \leq \min\left(\frac12,\frac{1}{\log D}\right)\right)
\tag{3}
]

of the functions (F(s)=\zeta(s,\chi)) (if (\chi) is not the principal character (\chi_0)) and (F(s)=\zeta(s,\chi_0)(s-1)/(s-2)), which is proved by applying function-

equation $\zeta(s,\chi)$, the theorem of Fragmén—Lindelöf and the theorem of Dech (${}^7$). If $\chi'=\chi_0$, then (2) is a consequence of the estimate
[
\operatorname*{Res}{s=1}\zeta(s,\chi_0)=D^{o(1)}\quad(D\to\infty),
]
which in turn follows from the estimate of the residue of the Dedekind zeta-function of the field $K$ ((${}^8$), (16)). For $\chi'\ne\chi_0$, (2) is proved by contour integration of the function
[
D^{a(s-1)}\Gamma(s-1)\zeta(s,\chi')\zeta(s,\chi_0)'
]
(where $a\ge 2n+1$), taking into account the fact that the coefficients $a_m$ of the expansion
[
\zeta(s,\chi')\zeta(s,\chi_0)=\sum\quad(\sigma>1)}^{\infty}a_m m^{-s
]
are nonnegative and $a_m\ge 1$ for
[
m=1^{2n},\,2^{2n},\,3^{2n},\ldots .
]

Lemma 2. For a suitable $A>0$ (depending only on $n$) and
[
\lambda_0=A\log\frac{eA}{\delta_0\log D},\qquad
\delta_0=
\begin{cases}
\delta, & \text{if }\delta\le A/\log D,\
A/\log D & \text{otherwise},
\end{cases}
]
the rectangle
[
(1-\lambda_0/\log D\le \sigma\le 1,\ |t|\le D)
]
contains no zeros $\rho\ne 1-\delta$ of the function $\prod_{\chi}\zeta(s,\chi)$.

Let
[
\rho_0=1-\lambda/\log D+it_0
]
be any zero of the function $\zeta(s,\chi)$ with $\lambda\le \log D$, $|t_0|\le D$. The following cases are considered separately:
[
1)\ \chi=\chi_0\ne\chi';\qquad
2)\ \chi\ne\chi_0\ne\chi';\qquad
3)\ \chi\ne\chi_0=\chi';\qquad
4)\ \chi=\chi_0=\chi'. \tag{4}
]
In the first three cases the following functions, respectively, are used:
[
f(s)=\zeta(s,\chi)\zeta(s-\delta,\chi\chi'),\quad
\zeta(s,\chi)\zeta(s+\delta,\chi\chi'),\quad
\zeta(s,\chi)
]
and the constructions of K. A. Rodosskii ((${}^2$), Ch. II) are repeated with minor changes. In the fourth case (4) one may proceed as follows. By the process of displacement (cf. (${}^2$), Ch. I, Lemma 4) a “convenient” zero $\rho_1=\beta_1+it_1$ of the function $\zeta(s,\chi_0)$ is found. If $|\tau_1|\ge 7$, then $f(s)=\zeta(s,\chi_0)$ is used and the arguments of the third case (4) are repeated. For $|\tau_1|<7$ the function
[
f(s)=\zeta(s,\chi_0)\zeta_1(s+\delta)G(s-\delta)
]
is used, where
[
\zeta_1(s)=\zeta(s)\prod_{p/N\mathfrak f}(1-p^{-s}),\qquad
G(s)=\zeta(s,\chi_0)/\zeta_1(s),
]
which is regular in the region $(\sigma>0,\ |t|<14)$ (where $\zeta(s)\ne 0$) and has the “convenient” zero $\rho_1+\delta$; analogous arguments are carried out with it.

Lemma 3. Let $N(\alpha,T)$ be the number of zeros of the function $\prod_{\chi}\zeta(s,\chi)$, belonging to the rectangle $(1-\alpha\le \sigma\le 1,\ |t|\le T)$. For a suitable $C>0$ (depending only on $n$) and all $\lambda\in[0,\log D]$ we have
[
N\left(\lambda/\log D,\ e^\lambda/\log D\right)<e^{C\lambda}.
]

Let $\nu(x,\mathfrak H)$ denote the number of ideals $\mathfrak a\in\mathfrak H$ with norm $N\mathfrak a\le x$. Using (3) and contour integration, we prove the estimate
[
\nu(x,\mathfrak H)=\gamma x+O\left(D^{2/3}x^{1-1/k}\right),
\qquad
\text{where }x\ge 1,\ k=\frac12(n+3),\quad
\gamma=h^{-1}\operatorname*{Res}_{s=1}\zeta(s,\chi_0), \tag{5}
]
$h$ denotes the number of classes $\mathfrak H$ ($h\le D$ according to (${}^4$), p. 66, and (${}^9$), § 3). Let
[
a_m\quad(m=1,2,\ldots,X); \tag{6}
]

such a collection of ideals of the field (K), for which

[
\sum_{\substack{a_m\ \mathfrak b_0/a_m}} 1 = X/f(\mathfrak b)+R_{\mathfrak b}
]

(where (\mathfrak b) is any ideal and (f(\mathfrak b)) is a multiplicative function (>0)), and let (N_z) ((z>1)) denote the number of those ideals (\mathfrak b) which are not divisible by any prime ideal (\mathfrak p) with norm (\le z), except for prime ideals (\mathfrak q) of some set (Q) (by which the ideals (a_m) may be divisible). Further, let (\mu(\mathfrak a)) be the Möbius function of ideals and

[
F(\mathfrak b)=\sum_{\mathfrak d/\mathfrak b}\mu(\mathfrak d)f\left(\frac{\mathfrak b}{\mathfrak d}\right),
\qquad
S_z=\sum_{\substack{\mathfrak b\ N\mathfrak b\le z\ \mathfrak q\nmid \mathfrak b}}
\frac{\mu^2(\mathfrak b)}{F(\mathfrak b)},
\qquad
S_z(\mathfrak i)=
\sum_{\substack{\mathfrak b\ (\mathfrak b,\mathfrak i)=\mathfrak o\ N\mathfrak b\le z/N\mathfrak i,\ \mathfrak q\nmid \mathfrak b^{\,i}}}
\frac{\mu^2(\mathfrak b)}{F(\mathfrak b)},
]

[
\lambda_{\mathfrak i}=
\begin{cases}
\mu(\mathfrak i)\displaystyle\prod_{\mathfrak p/\mathfrak i}(1-1/f(\mathfrak p))^{-1}S_z(\mathfrak i)/S_z,
& \text{if } \mathfrak q\nmid \mathfrak i,\ N\mathfrak i\le z,\[6pt]
0, & \text{otherwise.}
\end{cases}
]

((\mathfrak q\nmid \mathfrak b) means that (\mathfrak q) does not divide (\mathfrak b).)

By Selberg’s method ((^{10})) one proves the estimate

[
N_z\le X/S_z+
\sum_{\substack{\mathfrak i_1,\mathfrak i_2\
N\mathfrak i_1\le z,\ N\mathfrak i_2\le z\
\mathfrak q\nmid \mathfrak i_1\mathfrak i_2}}
\left|\lambda_{\mathfrak i_1}\lambda_{\mathfrak i_2}R_{\mathfrak i_1\mathfrak i_2/(\mathfrak i_1,\mathfrak i_2)}\right|.
\tag{7}
]

If in the special case (6) is the collection of all ideals (\mathfrak a) of the class (\mathfrak H) with (N\mathfrak a\le x), then, on the basis of (5),

[
f(\mathfrak b)=N\mathfrak b,\qquad
R_{\mathfrak b}\ll D^{3/2}(x/N\mathfrak b)^{1-1/k},
\qquad
S_z=\sum_{\in(z)}1/N\mathfrak b,
]

where ((z)) denotes the set of those ideals of the field (K) the product of the norms of all distinct prime ideals of which is (\le z). Let (\mathfrak g) run through all ideals of the field (K) divisible only by prime ideals (\mathfrak q\in Q), and let (V=\sum_{\mathfrak g}1/N\mathfrak g); then

[
S_z\cdot V\ge
\sum_{\mathfrak b\in(z)}1/N\mathfrak b
\ge
\sum_{\substack{\mathfrak b\ N\mathfrak b\le z\ (\mathfrak b,\mathfrak f)=\mathfrak o}}
1/N\mathfrak b.
]

For (x\ge D^{5k}) and (z\ge \max(x^{1/4},D^{3k})), the last sum is
(>c_2 h\gamma\log z>c_3 h\gamma\log x). If (\sum_{\mathfrak q}1/N\mathfrak q\ll 1), then (V\ll1) and (S_z>c_4h\gamma\log x), whence follows the estimate (\ll x/h\log x) for the main term in (7). Under the additional restriction
(z^{2/k}=x^{1/k}/D^{1/3}(h\gamma)^2h\log x\ge D^6), the same estimate can be proved for the remainder term in (7) (cf. ((^{11})), § 7), and under these conditions

[
N_z\ll x/h\log x.
\tag{8}
]

Let (\pi(x,\mathfrak H)) denote the number of prime ideals of the class (\mathfrak H) with norm (\le x), and (V(x;\mathfrak H,y)) the number of those ideals (\mathfrak a) of the class (\mathfrak H) with (N\mathfrak a\le x), all prime divisors (\mathfrak q) of which have norms in ([y,y^2]) ((1\le y\le\sqrt{x})). Simple consequences of (8) are the estimates

[
\pi(x,\mathfrak H)\ll x/h\log x,\qquad
V(x;\mathfrak H,y)\ll x/h\log x
\quad (x\ge D^{5k},\ 1\le y\le\sqrt{x}).
\tag{9}
]

In the proof of the first of them the empty set (Q) is used (cf. ((^{11})), § 8).

Using the estimates (9), the lemma is proved by arguments similar to those carried out by K. A. Rodosskii ((2), Ch. I), with only the important difference that the numbers (m,u,v) ((2), p. 339) are replaced by the norms of such ideals (\mathfrak a), all prime factors of which have norms belonging to some interval ([y,y^2]), where (D^{\log(1+\lambda)/(2+2\lambda)} \leqslant y < D^{6k}).

Theorem 1 is proved by the method set forth in (12) for the special case (n=1). Let (q) be a natural number (>1), and let (\chi_q) be a Dirichlet character (\bmod q). It can be shown that

[
\zeta(s,\chi_q,\chi)=\sum_{\mathfrak a}\frac{\chi_q(N\mathfrak a)\chi(\mathfrak a)}{N\mathfrak a^s}\quad(\sigma>1)
\tag{10}
]

coincides with a certain Hecke function with character modulo (\mathfrak f[q]). Using the functions (10), it is proved that if there exists an ideal (\mathfrak a\in\mathfrak H) with (N\mathfrak a\equiv l(\bmod q)), ((l,q)=1), then the assertion of Theorem 1 with (D=|\Delta|q^nN\mathfrak f) holds for primes (p\equiv l(\bmod q)).

Let henceforth (K) be a quadratic field generated by the square root (\sqrt{\Delta}), where (\Delta) is the fundamental discriminant, and let (\mathfrak f=[k]) with a natural number (k>1). Ideals (\mathfrak a,\mathfrak b) (having no common divisor with (k)) are, by definition, in one and the same class (\mathfrak C) if there exist integers (\alpha,\beta\in K), congruent modulo ([k]) to rational numbers having no common divisor with (k), such that (N\alpha\beta>0) and (\mathfrak a[\alpha]=\mathfrak b[\beta]). It can be shown that each class (\mathfrak C) is a sum of one and the same number (j\geqslant1) of classes (\mathfrak H). To each class (\mathfrak C) there corresponds uniquely a class (\mathfrak F) of primitive binary quadratic forms with discriminant (d=\Delta k^2) (if (d<0), considering only positive forms), and every form (\psi\in\mathfrak F) represents norms of ideals of the class (\mathfrak C) ((13), pp. 123–124). Let (X(\mathfrak C)) be the characters of the classes (\mathfrak C). The corresponding function (\zeta(s,X)) is represented by the finite sum

[
\zeta(s,X)=h^{-1}\sum_{\mathfrak C}X(\mathfrak C)\sum_{\mathfrak H\subset\mathfrak C}\sum_{\chi}\overline{\chi}(\mathfrak H)\,\zeta(s,\chi).
]

Using (3), we obtain estimates of (|\zeta(s,X)|), which make it possible to prove Lemmas 1–3 in an analogous way with (X) in place of (\chi). For real (X) the function

[
\prod_{\substack{\mathfrak p\ N\mathfrak p/\Delta,\;N\mathfrak p\nmid k}}
(1-X(\mathfrak p)N\mathfrak p^{-s})\zeta(s,X)
]

is represented as the product of two Dirichlet (L)-functions with characters (\bmod D) ((14), p. 67). This fact, together with Siegel’s theorem, permits replacing in (2) the exponent by an arbitrarily small positive constant (\varepsilon>0) (if (D>D_0(\varepsilon))). Using this circumstance, Theorem 2 is proved according to the scheme set forth in (3).

Latvian State University
named after Pēteris Stučka

Received
21 III 1961

References

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