7. On the Number of the Primes in an Arithmetic Progression.

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1 7. On the Number of the Primes in an Arithmetic Progression. By Tikao TATUZAW A. (Received Jan. 20, 1951) Let ƒè( 3) be a positive integer. We denote by ƒô one of the ƒó(ƒè) Dirichlet's characters with modulus ƒè, where ƒó(ƒè) is Euler's function. Let ƒä(s,w) and L(s,ƒÔ) denote the functions defined for ƒð>1 by ƒ n=0(n+w)-s and ƒ n=1ƒô(n)n-s respectively, where 0<w 1 and s=ƒð+it is a complex variable. ƒô is the complex conjugate character of ƒô. Throughout the paper we denote by c with index the absolute positive constant. We use the symbols Y=O(X) and Y áx for positive X when there exists a c0 satisfying Y c0x in the full domain under consideration. Let ƒô0 be the principal character. We know from Page's theorem [5] (Numbers in square brackets refer to the appended list of references) that of all the L-functions to modulus ƒè there is at most one function which has a zero in the region (1) Further if such a function exists, let us denote it by L(s,ƒÔ1), then the only zero of L(s,ƒÔ1) in the region (1) is a simple real zero ƒà1. We write Let ƒï(ƒô)=ƒï=ƒà+iƒá be a typical zero of L(s,ƒÔ) those for which ƒá=0,ƒà 0 and ƒï=ƒà1, 1-ƒÀ1, for ƒô=ƒô1 being excluded. The purpose of the present paper is to improve the order of the remainder in the asymptotic formula for ƒî(x;ƒè,l) the number of primes p satisfying p x, p ßl (mod. ƒè). The main result is that, if x exp (c2 logƒè loglogƒè), (ƒè,l) =1, then (2) where This is a slight extension of the result obtained by Rodosskii [7]. As an im mediate consequence of (2) we obtain, for x 3,

2 94 Tikao TATUZAWA where ƒî(x) denotes the number of primes not exceeding x. This is slightly better than the result obtained by Titchmarsh [10]. These results are based on the following theorems. Theorem 1. There exists a c5 such that all L(s,ƒÔ) to modulus ƒè have no zeros in the region except real axis. Theorem 2. If x T 2, then where ƒ (n) is logp if n is a positive power of a prime p, and is 0 otherwise. Theorem 3. Let N(ƒ,T) denote the total number of zeros of all L(s,ƒÔ) to modulus ƒè in the rectangle ƒ ƒð 1, t T. If 1/2 ƒ 1 and T 2, then (3) where c is a positive constant. Theorem 4. Let ƒõ(x;ƒè,l)=ƒ n xn ßl(mod,ƒÈ)ƒ (n). If x exp(c6logƒèloglogƒè), (ƒè,l)=1, then To prove Theorem 1, we use a result of Loo-Keng Hua [6] concerning Vino gradow's estimation of Weyl's sum. We follow in the wake of Titchmarsh [10]. For Theorem 2, I have sketched the proof in my previous paper [8]. We now proceed to fill in the details. Theorem 3 is the principal weapon of our argument for which I have given the whole proof in [8]. The constant c in (3) is determined by the estimation so we can take c=1/6. From Theorem 4 we can easily deduce the main theorem ([1], Lemma 10).

3 On the Number of the Primes in an Arithmetic Progression. 95 I express my hearty thanks to Prof. Suetuna and Mr. Iseki far their kind a dvices. 1. Proof of Theorem 1. Lemma 1. Let S=ƒ Px=1exp2ƒÎi(ƒ xn+1+ƒ nxn+ c+ƒ 1x+ƒ 0) where P is a positive integer and ƒ, ƒ n, c, ƒ 1, ƒ 0 are real numbers. If 2(n+1) a P 1 and n c8, then Proof. We write where Q is a positive integer to be determined later. Then (4) Let t be a positive integer. We have where Let ƒõ(nn, c,n1) be the number o f solutions of Then we have

4 96 Tikao TATUZAWA (5) First the expression (6) From Loo-Keng Hua's theorem ([6], Theorem 1) the right integral of (6) does not exceed (7) if 2t>1/4n(n+1)+ln. For this, we put (8) Then it is easily verified that (9) Next the expression (10) Now we assume that

5 On the Number of the Primes in an Arithmetic Progression. 97

19 1. K. Iseki: A divisor problem involving prime numbers; Jap. J. Math. 21 (1953). 2. E. Landau: Handbuch der Lehre von der Verteilung der Primzahlen, Ed. 1; Leipzig, Teubner, E. Landau: Uber einige Summen, die von den Nullstellen der Riemannschen Zetafunk tion abhangen; Acta Math., 35 (1912), E. Landau: Vorlesungen uber Zahlentheorie, Bd. 2; Leipzig, Hirzel, A. Page: On the number of the primes in an arithmetic progression; Proc. London Math. Soc., 39 (1935), Loo-Keng Hua: An improvement of Vinogradow's mean-value theorem and several ap plications; The Quarterly Journal of Math., Oxford Series, 20 (1949), K. A. Rodosskii: On the zeros of Dirichlet's L-functions; Izvestiya Akad. Nauk SSSR Ser. Mat., 13, (1949). 8. T. Tatuzawa: On the zeros of Dirichlet's L-functions; Proc. of the Japan Acad., 26 (1950), No. 9, N. G. Tchudakoff: On Goldbach-Vinogradow's Theorem; Annals of Math., 48 (1947). 10. E. C. Titchmarsh: On ƒä(s) and ƒî(x); The Quarterly Journal of Math., 9 (1938),

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Now we determine an by the following equations; 8. On a Power Series which has only Algebraic Singularities on its Convergence Circle. U. By Masatsugu TSUJI. (Received April 1, 1927.) 1. 1. Let f(x)= ƒ n=0anxn have the convergence radius 1, then Jentzch