Download PDF by Carlos J. Moreno: Advanced Analytic Number Theory, Part I: Ramification

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By Carlos J. Moreno

ISBN-10: 0821850156

ISBN-13: 9780821850152

ISBN-10: 6419835127

ISBN-13: 9786419835129

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Additional info for Advanced Analytic Number Theory, Part I: Ramification Theoretic Methods

Example text

Let / 3 > a > l be two given positive numbers and v > 1. Let 2Ji denote the set of integers n satisfying the following conditions: * » - # <«<;*», n^=0 (mod pi), l

Forh. Trondhjem, 39, 145—148. , 1952. On Elementary Methods in Prime Number Theory and Their Limitations, Den 11 - te Scand. Mat. Kongr. Trondheim, 13 — 22. 29 SCIENCE RECORD New Ser. Vol. I, No. 3, 1957 MATHEMATICS ON SIEVE METHODS AND SOME OF THEIR APPLICATIONS* WANG YUAN (£ 'It) Institute of Mathematics, Academia Sinica {Communicated by Prof. Hua, L. , Member of Academia Sinica) In 1919, V. BruntI] first proved the following result: For sufficiently large x, there exists, between x — x2 and x, an integer of at most 11 primes factors.

E. the respective numbers of prime factors of n and x — n are at most 3 and 4. Theorem 1 is proved. ). Then (17) becomes Pw(x,x1'i,x1'6) = l>rT-> z ^ o - J2 ^ log ^ p|n=>p>a;1/4 1 5 p|(n+2)=>p>z / Theorem 2 follows. A. Selberg announced that some results might be possibly obtained by his method, for example, (2,3) in [9] and (3,3) in [10]. However, the proofs of these results did not appeared in the literature till now. The present method can also be used to prove the following results which will be published in other papers.

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Advanced Analytic Number Theory, Part I: Ramification Theoretic Methods by Carlos J. Moreno


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