# Atle Selberg Collected Papers by Atle Selberg

By Atle Selberg

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In 3)1, the number of elements with at least m prime i i_ divisors in the interval x ® < p ^ x a is not exceeding -L ^ p(- - xT\ Proof. For l ^ y ^ i — s, let Fj be the subset of 2ft whose elements are divided by ps+j- Indeed, Fj is the set of integers n satisfying the following conditions: * » - * < » < x v , « # 0 ( m o d ^ , ) , l < i < s , n # 0 ( m o d pj+k), l < A < ^ - s , « = 0(mod ps+j). (5) 31 3 Evidently, the number of elements of Tj is not exceeding the number of integers satisfying the following conditions: ^ _4

1 (1922) 179-194. [2] H. Heilbronn, tiber die Verteilung der Primzahlen in Polynomen, Math. Ann. 104 (1931) 794-799. [3] G. Ricci, Su la congettura di Goldbach e la constante di Schnirelmann, Ann. della R. Scu. Normale Sup. di Pisa (2) 6 (1973) 70-115. [4] T. Nagell, Generalisation dun theoreme de Tchbycheff, J. Math. (8) 4 (1921) 343-356. [5] D. V. Widder, The Laplace Transform (Princeton Univ. Press, 1946), p. 192. [6] E. V. Chulanovski, Some estimations connected with Seiberg's new method, Dokl.

104 (1931) 794-799. [3] G. Ricci, Su la congettura di Goldbach e la constante di Schnirelmann, Ann. della R. Scu. Normale Sup. di Pisa (2) 6 (1973) 70-115. [4] T. Nagell, Generalisation dun theoreme de Tchbycheff, J. Math. (8) 4 (1921) 343-356. [5] D. V. Widder, The Laplace Transform (Princeton Univ. Press, 1946), p. 192. [6] E. V. Chulanovski, Some estimations connected with Seiberg's new method, Dokl. Akad. Nauk SSSR 43 (1948) 491-494. SCIENCE RECORD Vol. I, No. 1, 1957 MATHEMATICS ON SIEVE METHODS AND SOME OF THE RELATED PROBLEMS* WANG YUAN (3E X) Institute of Mathematics, Academia Sinica {Communicated by Prof.