On the distribution of \(\alpha p\) modulo one. II
Let \(\alpha,\beta\) be real numbers, with \(\alpha\) being irrational. Using exponential sums I.~M.~Vinogradov showed that there are infinitely many primes \(p\) such that \(\|\alpha p-\beta\|<p^{-\tau}\) provided that \(\tau<{1\over 5}\); here \(\|x\|\) is the minimum distance from \(x\) to an integer. More recently, \textit{G.~Harman} [J. Lond. Math. Soc., II. Ser. 27, 9-18 (1983; Zbl 0504.10018)] introduced the sieve method to tackle the problem, showing that \(\tau={3\over 10}\) is an admissible value. This was improved to \(\tau={4\over 13}\) by the author [J. Number Theory 45, 241-253 (1993; Zbl 0786.11042)], and then to \(\tau={7\over 22}\) by \textit{G. Harman} [Proc. Lond. Math. Soc., III. Ser. 72, 241-260 (1996; Zbl 0874.11052)]. In the paper being reviewed, the author shows that \(\tau={9\over 28}\) is admissible. By applying Buchstab's identity several times, the problem becomes one of estimating the `deficiencies' for various sums arising from the related sieve problem; roughly speaking a deficiency is an admissible coefficient associated with a term being attached to an asymptotic formula in order to give a valid lower bound for the sum concerned. Such deficiencies can be expressed as integrals involving the Buchstab function, and much tedious calculations have to be undertaken in order to deliver the claimed admissible exponent.
scientific article; zbMATH DE number 1542544