Proofs to one inequality conjecture for the non-integer part of a nonlinear differential form

We prove the conjecture for the non-integer part of a nonlinear differential form representing primes presented in (Lai in J. Inequal. Appl. 2015:Article ID 357, 2015) by using Tumura-Clunie type inequalities. Compared with the original proof, the new one is simpler and more easily understood. Similar problems can be treated with the same procedure.


Introduction
The non-integer part of linear and nonlinear differential forms representing primes has been considered by many scholars. Let [x] be the greatest non-integer not exceeding x. In , Danicic [] proved that if the diophantine inequality |λ  p  + λ  p  + λ  p  + η| < ε () satisfies certain conditions, and primes p i ≤ N (i = , , ), then the number of prime solutions (p  , p  , p  , p  ) of () is greater than CN  (log N) - , where C is a positive number independent of N . Based on the above result, Danicic [] proved that if λ, μ are non-zero real numbers, not both negative, λ is irrational, and m is a positive non-integer, then there exist infinitely many primes p and pairs of primes p  , p  and p  such that [λp  + μp  + μp  ] = mp.
In particular [λp  + μp  + μp  ] represents infinitely many primes. Brüdern et al. [] proved that if λ  , . . . , λ s are positive real numbers, λ  /λ  is irrational, all Dirichlet L-functions satisfy the Riemann hypothesis, s ≥   k + , then the non-integer parts of are prime infinitely often for natural numbers x j , where x j is a natural number.
Recently, Lai [] proved that, for non-integer r ≥  k- +  (k ≥ ), under certain conditions, there exist infinitely many primes p  , . . . , p r , p such that And he also conjectured that the above results are true when primes p j in (.) are replaced by natural numbers x j . In this paper we shall give an affirmative answer to this conjecture.

Main result
Our main aim is to investigate the non-integer part of a nonlinear differential form with non-integer variables and mixed powers ,  and . Using Tumura-Clunie type inequalities (see [, ]), we establish one result as follows.
Theorem . Let λ  , λ  , . . . , λ  be nonnegative real numbers, at least one of the ratios λ i /λ j ( ≤ i < j ≤ ) is rational. Then the non-integer parts of Remark It is easy to see by the differential from Theorem . that primes p j in (.) are replaced by a natural numbers x j and there exist infinitely many primes p  , . . . , p r and p such that [μ  p k  + · · · + μ r+ p k r+ ] = mp r , where m is a nonnegative non-integer (see []).

Outline of the proof
Throughout this paper, p denotes a prime number, and x j denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ  , λ  , . . . , λ  .
We write e(x) = exp(πix). We take X to be the basic parameter, a large real non-integer.
Since at least one of the ratios λ i /λ j ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ  /λ  is irrational. For the other cases, the only difference is in the following intermediate region, and we may deal with the same method in Section .
Since λ  /λ  is irrational, there are infinitely many pairs of non-integers q, a with |λ  /λ a/q| ≥ q - , (p, q) = , q >  and a = . We choose p to be large in terms of λ  , λ  , . . . , λ  , and make the following definitions. Put Next we estimate J. As usual, we split the range of the infinite integration into three sec- has infinitely many solutions in positive non-integers x  , x  , . . . , x  and prime p. From (.) we have which gives The proof of Theorem . is complete.

The neighborhood of the origin
where C is a positive constant and ρ = β + iγ is a typical zero of the Riemann zeta function. Then we have Proof It is obvious that The proofs of the other cases are similar, so we complete the proof of Lemma ..

Lemma . The following inequality holds:
Proof For α = , i = , , , , j = , , , k = , , we know that Lemma . The following inequality holds: Then we have we obtain Then we complete the proof of this lemma.
The proofs of the others are similar. So we omit them here.
Lemma . For every real number α ∈ D, we have Proof For α ∈ D and i = , , , , we choose a i , q i such that with (a i , q i ) =  and  ≤ q i ≤ Q. We note that a  a  a  a  = . If q  , q  ≤ P, then We recall that q was chosen as the denominator of a convergent to the continued fraction for λ  /λ  . Thus, by Legendre's law of best approximation, we have |q λ  λ a | >  q for all non-integers a , q with  ≤ q < q, thus On the other hand, which is a contradiction. And so for at least one i, P < q i Q. Hence we see that the desired inequality for W (α) follows from Weyl's inequality (see [], Lemma .).

The trivial region
for any A > .
The following inequality holds.

Conclusions
In this paper, we proved the conjecture for the non-integer part of a nonlinear differential form representing primes presented in [] by using Tumura-Clunie type inequalities. Compared with the original proof, the new one is simpler and more easily understood. Similar problems can be treated with the same procedure.