A note on Diophantine approximation by unlike powers of primes

Abstract

It is proved that if [Formula: see text] are nonzero real numbers, not all of the same sign and [Formula: see text] is irrational, then for given real numbers [Formula: see text] and [Formula: see text], [Formula: see text], the inequality [Formula: see text] has infinitely many solutions in prime variables [Formula: see text]. This result constitutes an improvement upon that of Liu for the range [Formula: see text].

A Diophantine Problem with Unlike Powers of Primes

“…e proof of (19) is quite similar to that given in Section 3 in [20]. For completeness of exposition, we briefly present the proof procedure below.…”

“…Subsequently, Liu [19] obtained 0 < σ(5) < 5/288. In [20], the first author and Qu showed that 0 < σ(5) < 5/252 is acceptable. Very recently, this result was improved by Zhu [21], who obtained 0 < σ(5) < 1/48.…”

On Diophantine approximation by unlike powers of primes

“…In this section, we give a low bound for the integral on the major arc M. First, we consider the standard major arc M * = {α : |α| ≤ X − + / −ε }. Using the idea due to Harman [7], we get the following lemma (one can also see section 3 of Mu and Qu [16]). One may improve the standard major arc to {α : |α| ≤ X − + / −ε } by using some ideas due to Languasco and Zaccagnini [10] (one can also see [5]).…”

“…Later, Mu [15], Liu [13], Mu and Qu [16] replaced in (1.1) with / , / and / respectively. In this paper, under some extra conditions of λ j , we get the following result.…”

“…In [13], Liu used S (λ α) P − / +ε . In [16], Mu and Qu used sieve method of Harman [7], and got S * (λ α) P − / +ε . Using the method of Mu and Qu [16], even if one got the best estimation S * (λ α) P − / +ε , / can only be replaced by / .…”