Article contents
On the distribution of √p modulo one
Published online by Cambridge University Press: 26 February 2010
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
- Copyright
- Copyright © University College London 1983
References
- 11
- Cited by