General SectionOn a Diophantine inequality over primes
Section snippets
Introduction and main result
In 1952, Piatetski-Shapiro [16] considered the following analogue of the Waring–Goldbach problem. Assume that is not an integer and let ε be a positive number. If r is a sufficiently large integer (depending only on c), then the inequality has a solution in prime numbers for sufficiently large N. More precisely, if the least r such that (1.1) has a solution in prime numbers for every and is denoted by , then, in [16], Piatetski-Shapiro proved
Notation
Throughout this paper, let , with or without subscripts, always denote prime numbers; η always denotes an arbitrary small positive constant, which may not be the same at different occurrences; means . As usual, ; means that ; means that . N is a sufficiently large real number; c is a fixed real number satisfying ; means that for all pairs for which it
Outline of the method
In this section, we shall introduce the idea of the proof of Theorem 1.1. First, we need the following lemma. Lemma 3.1 Let be real numbers, , and let ℓ be a positive integer. There exists a function which is k times continuously differentiable and such that and its Fourier transform satisfies the inequality Proof See Piatetski-Shapiro [16] or Segal [18]. □
We write
Preliminary lemmas
Throughout this paper, we denote by the function from Lemma 3.1 with parameters , and . Also, we denote by the Fourier transform of .
Lemma 4.1 For , we have Proof See Lemma 2.3 of Mu [15]. □
Lemma 4.2 Define Then for , there holds Proof See Lemma 2.5 of Mu [15]. □
Lemma 4.3 Let . Then we have Proof See
Exponential sums
In this section, we shall prove the exponential sum estimates which we will need to get the asymptotic formula (3.5). Lemma 5.1 Let α and β be real, , . Then we have Proof See Theorem 9 of Sargos and Wu [17]. □
Lemma 5.2 Let be a subdomain of the rectangle such that any line parallel to any
Asymptotic formulas
In this section, let be the Fourier transform of the function defined in Lemma 3.1. Define and
Lemma 6.1 Suppose that . Let be the continuous solution of the differential–difference equation Then for any , we have In particular, we have
Proof of Theorem 1.1
We start with (3.11). First, we shall establish the connections among and . By (3.1), (6.26) and (6.27), we have By prime number theorem, it is easy to get and From (7.1) and (7.2), we derive that
Acknowledgment
The authors would like to express the most sincere gratitude to Professor Wenguang Zhai for his valuable advices and constant encouragement.
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