1. Generalized Prime Number Theorem. By Takayoshi MITSUI (ReceivedJanuary 31, 1957) Let K be an algebraic number field of degree n, K(1),•c, K(ƒÁ1) the real conjugate fields of K, and K(ƒÁ1+1),•c, K(n) (n=ƒÁ1+2ƒÁ2) the complex conjugate fields of K where K(p+ƒÁ2)=(p=ƒÁ1+1,•c,ƒÁ1+ƒÁ2), we denote by P the direct product of ƒÁ1 real lines and ƒÁ2 complex planes, Let ƒÊ be an ideal number of K , and ƒÊ(1),•c,ƒÊ(n) the conjugates of ƒÊ as defined by Hecke [2] , Then V(ƒÊ)=(„ ƒÊ(1)„ ,•c,„ ƒÊ(ƒÁ1)„ ,ƒÊ(ƒÁ1+1),•c, ƒÊ(ƒÁ1+ƒÁ2) can be regarded as a point in P, and it is a problem to compute the number of prime ideal numbers ƒÖ such that V(ƒÖ) are in a certain domain in P. In case n+ƒÁ1=2, Rademacher [7] obtained the following result; let K be a real quadratic number field, Y1, Y2 positive number, a an ideal of K, ƒÏ a totally positive integer of K, (ƒÏ, a) t, If we denote by ƒÎ(ƒÏ, a; Y1, Y2) the number of prime numbers w satisfying the conditions (ƒÖ=ƒÏ(mod a), 0<ƒÖ(1)•…Y1, 0<ƒÖ(2)•…Y2, then we have (1) where h is the number of ideal classes, is a totally positive fundamental unit such that ƒÅ>1. On the other hand, it is known (Siegel [8], walfisz •m10]) that we have (2) where the remainder term is uniform with respect to the modulus k, that is , the constants in the remainder term do not depend on k under the assumption k•…(log x) A for a fixed positive number A. In the present paper, we shall obtain a result including all above theorems as special cases. Our Main Theorem will be stated and proved in our final paragraph •˜ 4, after some preparations in •˜•˜1-3. we begin by generalizing (2) to the case of arbitrary algebraic number field in •˜ 1. The aim of this paragraph will be attained by the theorem on ƒÎ(x, _??_), the number of prime ideals p in the class _??_ with Np•…x, given at the end of •˜1. Now ƒÎ(x, _??_) may be considered as a special case for ƒÉ=1 of a certain sum (see (3.52)) for Grossencharacter A of K. The estimation of this sum‡”*ƒÉ(ƒÖ) for the case ƒÉ=1 will be given as Lemma 8 in •˜2, after seven lemmas concerning the properties of ƒÄ-f unction with Grossencharacter, In •˜2, we shall prove three more lemmas needed to establish the uniformity of the remainder term of our final for 2 Takayoshi MITSUI mula with respect to the modulus. In •˜3, we shall generalize the following theorem of Rademacher [6] by means of which he could establish his above cited result (1); let K be a real quadratic field, a an ideal, ƒÅa a totally positive fundamental unit mod a such that and define a function W(ƒÊ) for ƒÊ=K as follows; If we take a totally positive integer o prime to a and denote by ƒÎ(a,ƒÏ; x, v) (0<v •…1) the number of totally positive prime numbers ƒÖ of K satisfying the conditions; ƒÖ=p (mod a), NƒÖ•…x, 0•…W(ƒÖ)<v, then we have The theorem, which will be the aim of this paragraph, will be formulated at the beginning. All results of •˜•˜1-2 are utilized in the proof of this theorem, by which our Main Theorem will be then proved in •˜4. The author wishes to express his cordial thanks to Professor S. Iyanaga for his encouragement during the preparation of this paper. 1. Improvement of prime ideal theorem. We consider Hecke's L-function L(s, x) of an algebraic field K with degree n, where x is a character mod a (a means the product of an ideal a and some infinite prime ideals p(i)•‡). We write as usual s=ƒÐ+it, ƒÐ=k(s), t=k(s) and denote by x0 the principal character and by A1, A2,•c positive absolute constants. (For details on the properties of L-functions, cf. Landau [3]). LEMMA 1. If x is a character mod a, then we have in the strip -1/2•…ƒÐ•…4 (1.1) „ L(s, x)(s-1)E(x)„ <A1(Na(1+„ t„ ))A2, where PROOF. It is obvious that L(s, x)(s-1)E(x) is regular in the strip -1/2•…ƒÐ•…4. First assume that x is a primitive character. On the line ƒÐ=4, we have „ L(s, x)(s-1)E(x)„ •…A3(1+„ t„ ), and on the line ƒÐ=-1/2, we have by the functional equation of L(s, x), „IL(s, x)(s-1)E(x)„ •…A4Na(1+„ t„ )n+1 Finally in the strip -1/2•…ƒÐ<4, we have„ L(s, x)(s-1)E(x)„ •…C1ec2t t, where C1, C2 are positive constants independent of t, ƒÐ. Hence by theorem of Phragmen-Lindelof, we have Generalized Prime Number Theorem. 3 (1.2) „ L(s, x)(s-1)E(x) „ •…A5Na(1+„ t„ )n+1. Next assume x is not primitive and let x0 be the primitive character induced from x. Then we have L(s, x)=L(s, x0) II (1-x0(p)Np-s), in which II(1-x0(p)Np-s)„ •…II(1+Np1/2)•…A6IINp•…A6Na. Therefore (1.1) is obtained. LEMMA 2. In the region „ t„ •†4,ƒÐ•†1-A7/log(„ t„ Na), we have L(s, x) •‚0. PROOF. 1t suffices to prove for t•†4. Suppose that ƒÀ+iƒÁ(ƒÁ•†4) is a zero point of L(s, x). Taking ƒÐ0=1+A/log(ƒÁNa) (A>0 will be determined later) we denote ƒÐ0+iƒÁ by s0. Then In the circle „ s-s0„ •…1/2, we have by Lemma 1 Therefore by lemma ƒÀ of Titchmarsh [9] p. 49, We obtain analogously Taking A sufficiently small, we have Hence, by a well-known inequality we obtain which gives Since we may take A small, the value of the formula in the brackets is surely positive, that is, Thus lemma is proved. LEMMA 3. (Titchmarsh) Suppose that f(s) is regular in the circle „ s-s0„ •… 4 Takayoshi MITSUI and „ f (s)/f(s0)„ <eM (M>1), „ f•L(s0)/f(s0)„ <M/r, further f(s)•‚0 in the domain defined by the conditions „ s-s0„ •…r and ƒÐ•†ƒÐ0-2ƒÁ•L (0<ƒÁ•L <ƒÁ/4). Then we have („ s-s0„ •…r). PROOF. See Titchmarsh [9] p. 50. LEMMA 4. In the region „ t„ •†5, ƒÐ•† 1-A7/8 log(„ t„ Na), we have (1.3) PROOF. Again we may assume t•†5. Let ƒÁ•†5 and put s0=ƒÐ0+iƒÁ. Then L(s, x) is regular in the circle „ s-s0„ •…1/2 and by Lemma 2 L(s, x)•‚0 in the domain defined by the conditions „ s-s0„ •…1/2 and ƒÐ•†ƒÐ0-2ƒÁ•L. Further we have in the circle„ s-s0„ •…1/2 and Therefore, putting A19=max(A17, A18/2), M=A19log(ƒÁNa), ƒÁ=1/2, we have by Lemma 3 („ s-s0„ •…ƒÁ). (1.3) follows at once. LEMMA 5. If x is a complex character mod a, then in the region „ t„ •…10, ƒÐ•†1 -A21/log(Na+1), L(s, x) has no zero point. PROOF. This is similar to the proof of Lemma 2, so we may omit it. LEMMA 6. If x is a real character mod a and x•‚x0, then L(s, x)•‚0 in the region; PROOF. Suppose that lemma is not true, then for any small positive number c there exists a zero point ƒÀ+iƒÁ of L(s, x) such that Now let a be a positive number such that 1•†((a+1)2+4a2)(a+1/2) and put ƒÐ0=1 +c/a log(Na+1), s0=ƒÐ0+iƒÁ. Taking c small, we may assume that two zero points ƒÀ+ iƒÁ, ƒÀ-iƒÁ of L(s, x) are in the circle „ s-s0„ •…1/4. In the circle „ s-s0„ •…1/2, L(s, x) is regular and „ L(s, x)„ <(Na+1)A24, so we have Generalized Prime Number Theorem. 5 („ s-s0„ •…1/2). Therefore by the lemma a of Titchmarsh [9] p. 49 (1.4) where ƒÏ runs through the zero points of L(s, x) in the circle „ s-s0•…1/4, so we have clearly Again taking c sufficiently small, we have so we have from (1.4) Further taking e small, This and the inequality give (1.5) This is a contradiction and proof is completed. LEMMA 7. If x is a real character mod a and x•‚x0, then L(s, x)•‚0 in the region 0<„ t„ •…6, ƒÐ•†1-A28/log(Na+1). PROOF. Let ƒÀ+iƒÁ (0<ƒÁ•…6) be a zero point of L(s, x). Then by Lemma 6, we may assume A22/log(Na+1)•…ƒÁ•…6. Put ƒÐ0=1+A/log(Na+1) (A>0) and consider (1.6) First taking A sufficiently small, we have Next in the circle „ s-s0„ •…1/2 (s0=ƒÐ0+iƒÁ) so we have Finally putting s0=ƒÐ0+2iƒÁ, we obtain 6 Takayoshi MITSUI in the circle •bs0-s•b•…1/2, hence by lemma ƒÀ of Titchmarsh [9] p. 49 Since A may be arbitrarily small, we may assume that A/(A2+A222)<1/4, that is, that the inequality holds.