Acta Scientiarum Universitatis Pekiniensis (Naturalum)

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[i] Jneunr, 10. B., Mamen. CB. 15:57, 1—11, 1944,

[2] Jneenr, 10. B., Mamem. CB. 15: 57, 1389—178, 1944.

[3] Jen, 10. B., Mamen. CB. 15: 57, 347—368, 1944.

[4] Pogocernfi. K. A., Mamen. CB. 34 (76): 2, 331—356, 1954.

[5] Paul. Turan. Eine neue methode in der analysis und deren anweendun gen. [6] GRE ASat, 1:5, 283-285, 1957.

[7] Uyazanoz., Beedenues Teopusw L-Oyuxyut dupurse.

[8] Pogoccxnit. KR. A. H3e. THCCCP. Cepua Mame. 18:4, 315—328, 1949. [9] Selberg. A., Norske Vid Trondhjem. 19:18, 64—97, 1947.

[10] Busorpajor, HU. M. Ocnosw meopuu Yuce..

[11] Titchmarsh. E. C., The Theory of the Riemann Zeta-Function. 151. [12] fH (7. 145—167.

[13] Pojoccxuit. KH. A., Mamem. CB, 86 78:2, 341—348, 1955.

[14] Landau. E., Handbuch der Lehre von der Verteilung der Primzahlen. [15] wee [14].

ON THE LEAST PRIME IN AN ARITHMETICAL PROGRESSION Pan Oheng-tung (Department of Mathematics and Mechanics)

Apsrracr Let Poin(D,1) denote the least prime in an arithmetical progression nD+1, with 1</<D—1 and (/,D)=1, then on the grand Riemann hypothesis for sufficiently large D, we may prove very easily that Pe Ot) <2 (1) where € is any positive number. In 1944, 10. B. Jimmi 24°33 proved without any hypothesis the existence of a positive absolute constant A, such that Pain (D, 1) <D*. (2) But his proof covered more than sixty pages. In 1954, lh. A. Pogoccnnit gave a simpler proof of (2), but P. Turan remarked at the end of his book that Pogzocenmi’s proof gives no information of the definite value of A and suggested to determine the constant by his own method. We have not, however, seen any paper written according to this suggestion. In this paper, we shall give a proof of the following theorem. Theorem. For sufficiently large D, we have Poin (Di) <D“, (3) where A, <5,448.