## The Annals of Probability

### A Log Log Improvement to the Riemann Hypothesis for the Hawkins Random Sieve

C. C. Heyde

#### Abstract

This paper is concerned with the Hawkins random sieve which is a probabilistic analogue of the sieve of Eratosthenes. Analogues of the prime number theorem, Mertens' theorem and the Riemann hypothesis have previously been established for the Hawkins sieve. In the present paper we give a more delicate analysis using iterated logarithm results for both martingales and tail sums of martingale differences to deduce a considerably improved $\log\log$ replacement for the Riemann hypothesis result.

#### Article information

Source
Ann. Probab., Volume 6, Number 5 (1978), 870-875.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995433

Digital Object Identifier
doi:10.1214/aop/1176995433

Mathematical Reviews number (MathSciNet)
MR503956

Zentralblatt MATH identifier
0414.60032

JSTOR