Fractional Parts of Non-Integer Powers of Primes. II

Abstract

We continue to study the distribution of prime numbers p, satisfying the condition $\{p^{\alpha} \} \in I \subset [0; 1)$, in arithmetic progressions. In this paper, we prove an analogue of Bombieri–Vinogradov theorem for 0 < α < 1/9 with the level of distribution $\theta = 2/5 - (3/5) \alpha$, which improves the previous result corresponding to $\theta \leqslant 1/3$.

Fractional Parts of Noninteger Powers of Primes

“…In this work we only focus on the proof of Theorem 1. The main difference between this proof and the proof of Theorem 1 from [16] is the application of Heath-Brown identity [17] in place of Vaughan identity [24,Ch. 13].…”

“…As in the previous work [16] let us denote by E ⊂ N the subsequence of natural numbers n ∈ N : {n α } ∈ I , where α > 0 is any fixed non-integer, I is any subinterval of [0; 1). The distribution of primes from E was studied by a number of authors including Vinogradov, Linnik, Kaufman, Gritsenko, Balog, Harman, Tolev and many others (see [1]- [15]).…”

“…Later Gritsenko and Zinchenko [15] extended this result to all 1/2 α < 1 and θ 1/3. In [16] the author further extended it to all α > 0, α / ∈ N and θ 1/3. In the present paper we improve this result for small α, namely we show (1) for all θ 2/5 − (3/5)α, which goes beyond the range θ 1/3 if 0 < α < 1/9.…”

“…The way one deduces corollaries 1 and 2 from Theorem 1 is explicated in the Section 2 of [16]. In this work we only focus on the proof of Theorem 1.…”

“…The estimation of type I and type II sums is very similar to the one in [16], and it only requires the classical van der Corput second derivative test [23, Ch. 1, Theorem 5] due to the small size of α.…”