Abstract
In several papers Antal Balog, Glyn Harman and the author studied the distribution of p l (mod 1), where l 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 t be given. For what fixed positive real numbers t is it possible to prove an asymptotic estimate, as N→∞, for the number of primes p ≤; N satisfying { p l -t} -t ? In the present paper we deal with the weaker problem for what real numbers t > 0 an asymptotic estimate of this kind holds true for almost all t. When l < 1/2 we get a wider t-range for almost all t than it is hitherto possible to obtain for a single t.
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