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On the distribution of √p modulo one

Published online by Cambridge University Press:  26 February 2010

Glyn Harman
Glyn Harman
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2BZ
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Extract

It was shown by Vinogradov (see Theorem 7, Chapter 4 of [18]) that, for ε > 0, there are infinitely many solutions in primes p of the inequality

where {x} denotes the fractional part of x and y = 0.1. The value of γ was improved to

by Kaufman [9]. On the Riemann Hypothesis he showed that one can take γ =¼ The method used actually shows that, for any real β and any δ with 0 < 8 > 1, the number of primes p ≤ x satisfying

is

where π(x) denotes the number of primes not exceeding x. The sequence √p is, of course, a subsequence of the sequence n½ whose distribution modulo one has also been investigated (see Chapter 2, Section 3 of [10]: the argument for sequences nσ (0 < σ < 1) is entirely elementary). It is useful in this context to define the discrepancy (modulo one) of a sequence an by

Type
Research Article
Information
Mathematika , Volume 30 , Issue 1 , June 1983 , pp. 104 - 116
Copyright
Copyright © University College London 1983

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