A general inequality for conformally flat submanifolds and its applications

Author： Chen Bang-Yen  

Publisher： Springer Publishing Company

ISSN： 0236-5294

Source： Acta Mathematica Hungarica, Vol.106, Iss.3, 2005-01, pp. : 239-252

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Abstract

In several papers Antal Balog, Glyn Harman and the author studied the distribution of p&lgr; (mod 1), where &lgr; is a given real number lying in the interval (0,1) and p runs over the prime numbers. One of the main questions in these papers can be formulated in the following way: Let a real &thetas; be given. For what fixed positive real numbers &tgr; is it possible to prove an asymptotic estimate, as N→∞, for the number of primes p ≤; N satisfying {p&lgr;-&thetas;} < p-&tgr;? In the present paper we deal with the weaker problem for what real numbers &tgr; > 0 an asymptotic estimate of this kind holds true for almost all &thetas;. When &lgr; < 1/2 we get a wider &tgr;-range for almost all &thetas; than it is hitherto possible to obtain for a single &thetas;.

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