The integer part of a nonlinear form with integer variables

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R E S E A R C H

Open Access

The integer part of a nonlinear form with

integer variables

Kai Lai

*

*_{Correspondence:}

[email protected] College of Computer and Information Engineering, Henan University of Economics and Law, Zhengzhou, 450011, P.R. China

Abstract

Using the Davenport-Heilbronn method, we show that if

λ

1,

λ

2,. . .,

λ

9are positive

real numbers, at least one of the ratios

λ

i/

λ

j(1≤i<j≤9) is irrational, then the integer parts of

λ

1x₁3+

λ

2x3₂+

λ

3x₃4+

λ

4x4₄+

λ

5x₅5+· · ·+

λ

9x5₉are prime inﬁnitely often for natural numbersx1,x2,. . .,x9.

Keywords: Davenport-Heilbronn method; integer variables; diophantine approximation

1 Introduction

In , Brüdernet al.[] proved that ifλ, . . . ,λsare positive real numbers,λ/λis irra-tional, all DirichletL-functions satisfy the Riemann hypothesiss≥_k+ , then the integer parts of

λxk+λxk+· · ·+λsxks

are prime inﬁnitely often for natural numbersxj.

Motivated by [], using the Davenport-Heilbronn method, we consider the integer part of a nonlinear form with integer variables and mixed powers ,  and , and establish one result as follows.

Theorem . Letλ,λ, . . . ,λbe positive real numbers,at least one of the ratiosλi/λj(≤ i<j≤)is irrational.Then the integer parts of

λx+λx+λx+λx+λx+· · ·+λx

are prime inﬁnitely often for natural numbers x,x, . . . ,x.

It is noted that Theorem . holds without the Riemann hypothesis.

2 Notation

Throughout, we usepto denote a prime number andxjto denote a natural number. We

denote byδa suﬃciently small positive number and byεan arbitrarily small positive num-ber. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend

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onλ,λ, . . . ,λ. We writee(x) =exp(πix). We use [x] to denote the integer part of real variablex. We takeXto be the basic parameter, a large real integer. Since at least one of the ratiosλi/λj(≤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 .

Sinceλ/λis irrational, then there are inﬁnitely many pairs of integersq,awith|λ/λ–

a/q| ≤q–_{, (}_a_,_q_{) = ,}_q_{>  and}_a_{= . We choose}_q_{to be large in terms of}_λ_,_λ_{, . . . ,}_λ and make the following deﬁnitions.

NX, L=logN, N–δ=q, τ=N–+δ,

Q=|λ|–+|λ|–N–δ, P=Nδ, T=N_.

Letνbe a positive real number, we deﬁne

Kν(α) =ν

sinπ να π να



, α= , Kν() =ν, (.)

Fi(α) =

≤x≤X

eαx, i= , ,

Fj(α) =

≤x≤X

eαx, j= , ,

Fk(α) =

≤x≤X

 

eαx, k= , . . . , ,

G(α) =

p≤N

(logp)e(αp),

fi(α) = X



eαxdx, i= , ,

fj(α) = X



eαxdx, j= , ,

fk(α) = X



eαxdx, k= , . . . , ,

g(α) =

N



e(αx)dx.

It follows from (.) that

Kν(α)minν,ν–|α|–, (.) +∞

–∞ e(

αy)Kν(α)dα=max,  –ν–|y|. (.)

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J=: +∞

–∞ 

i=

Fi(λiα)G(–α)e

–

α

K (α)dα

≤logN

|λx+λx+λx+λx+λx+···+λx–p–|< ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N



=: (logN)N(X),

thus

N(X)≥(logN)–J.

To estimateJ, we split the range of inﬁnite integration into three sections, traditional named the neighborhood of the originC={α∈R:|α| ≤τ}, the intermediate regionD= {α∈R:τ<|α| ≤P}and the trivial regionc={α∈R:|α|>P}.

3 The neighborhood of the origin

Lemma . Ifα=a/q+β,where(a,q) = ,then

≤x≤N/t

eαxt=q– q

m=

eamt/q N/t



eβytdy+Oq/+ε +N|β|.

Proof This is Theorem . of [].

If|α| ∈C, by Lemma ., takinga= ,q= , then

Fi(α) =fi(α) +O

Xδ, i= , , . . . , . (.)

Lemma . Letρ=β+iγ be a typical zero of the Riemann zeta function,C be a positive constant,

I(α) = |γ|≤T,β≥

n≤N

nρ–e(nα), J(α) =O +|α|NN_LC_,

then

G(α) =g(α) –I(α) +J(α), (.) 



–

I(α)dαNexp–L_, _(.)

τ

–τ

J(α)dαNexp–L_. _(.)

Proof Equations (.), (.), (.) can be seen from Lemma , () and () given by

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Lemma . We have

 

– fi(α)



dαX–_, _i_{= , ,} 



–_

_f_j(α)dαX–_, _j_{= , ,}

 

–_ fk(α)



dαX–_, _k_{= , . . . , .}

Proof These results are from Lemma  of [].

Lemma . We have

C



i=

Fi(λiα)G(–α) –



i=

fi(λiα)g(–α)

K

(α)dαX  _L–_.

Proof It is obvious that

Fi(λiα)X



_, _f_i₍_λ_i_α₎_X_, _i_{= , ,}

Fj(λjα)X

 _, _f

j(λjα)X



_, _j_{= , ,}

Fk(λkα)X



_, _f_k₍_λ_k_α₎_X_, _k_{= , . . . , ,}

G(–α)N, g(–α)N, 

i=

Fi(λiα)G(–α) –



i=

fi(λiα)g(–α)

=F(λα) –f(λα) 

i=

Fi(λiα)G(–α)

+F(λα) –f(λα) 

i=

i=

Fi(λiα)G(–α) +· · ·

+F(λα) –f(λα) 

i=

fi(λiα)G(–α) +



i=

fi(λiα)

G(–α) –g(–α).

Then by (.), Lemmas . and ., we have

C

F(λα) –f(λα) 

i=

Fi(λiα)G(–α)

K

(α)dαN

–+δ_Xδ_X__N_X_+δ_,

C



i=

fi(λiα)

G(–α) –g(–α) K

(α)dα X

C

f(λα) 

K (α)dα

 

C

J(–α) –I(–α)K (α)dα

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X  

–_

f(λα)dα 



C

J(α)dα+  

–_

I(α)dα

 

X_X–_N_exp_–_L  

X_L–_.

The other cases are similar, and the proof of Lemma . is completed.

Lemma . We have

|α|>N–+δ

 i=

fi(λiα)g(–α)

K

(α)dαX 

– 

δ_.

Proof It follows from Vaughan [] that forα= ,

fi(λiα) |α|–



_, _i_{= , ,} _f_j₍_λ_j_α₎ |_α|–_, _j_{= , ,}

fk(λkα) |α|–



_, _k_{= , . . . , ,} _g_(–_α₎ |_α|–_. Thus

|α|>N–+δ

 i=

fi(λiα)g(–α)

K

(α)dα _|_α_| >N–+δ|

α|– _d_α_X–δ_.

Lemma . We have

+∞

–∞ 

i=

fi(λiα)g(–α)e

–

α

K

(α)dα X  _.

Proof From (.) one has

+∞

–∞ 

i=

–

α

K (α)dα

=

X



X



X



X



X



· · · X    N  +∞

–∞ e

α

λx+λx+λx+λx

+λx+· · ·+λx–x–  

K

(α)dαdx dx· · ·dxdxdxdxdx = 

,

X



· · · X  N  +∞ –∞ x–    x –  x –  x –  x –_  · · ·x

–_  e α _ i=

λixi–x–

 

·K

(α)dαdx dx· · ·dx = 

,

X



· · · X  N  x–    x –_  x –  x –  x –_  · · ·x

–_  ·max , –  i=

λixi–x–

 

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Let|_i_=λixi–x–_| ≤_, then



i=λixi–_≤x≤



i=λixi–_. Based on



i=

λixi–

 > ,



i=

λixi–

 <N,

one may take

λjX





i=

λi

–

≤xj≤λjX





i=

λi

–

, j= , . . . , ,

hence

+∞

–∞ 

i=

–

α

K

(α)dα≥  ,,



j=

λj





i=

λi

–

X_.

This completes the proof of Lemma ..

4 The intermediate region

Lemma . We have

+∞

–∞

_F_i₍λiα)



K

(α)dαX 

+ε_, _i_{= , ,} _(.)

+∞

–∞

Fj(λjα)



K

(α)dαX +_ε

, j= , , (.)

+∞

–∞

Fk(λkα)



K

(α)dαX 

+ε_, _k_{= , . . . , ,} _(.)

+∞

–∞

G(–α)K

(α)dαNL. (.)

Proof By (.) and Hua’s inequality, fori= , , we have

+∞

–∞

Fi(λiα)



K (α)dα

+∞

m=–∞

m+

m

Fi(λiα)



K (α)dα



m=

m+

m

Fi(λiα)dα+

+∞

m=

m– m+

m

Fi(λiα)dα

X+ε₊_X+ε +∞

m=

m–

X+ε_.

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Lemma . Suppose that(a,q) = ,|α–a/q| ≤q–_,_φ₍_x_{) =}_α_xk₊_α

xk–+· · ·+αk–x+αk, then

M

x=

eφ(x)M+εq–+M–+qM–k–k.

Proof This is Lemma . (Weyl’s inequality) of Vaughan [].

Lemma . For every real numberα∈D,let W(α) =min(|F(λα)|,|F(λα)|),then

W(α)X–δ+ε_.

Proof Forα∈Dandi= , , we chooseai,qisuch that

|λiα–ai/qi| ≤q–i Q– (.)

with (ai,qi) =  and ≤qi≤Q.

Firstly, we note thataa= . Secondly, ifq,q≤P, then

aq

λ

λ –aq

≤a/q

λα

qq

λα–

a

q

+a/q

λα

qq

λα–

a

q

PQ–<  q.

We recall thatqwas chosen as the denominator of a convergent to the continued frac-tion forλ/λ. Thus, by Legendre’s law of best approximation, we have|q_λλ–a|>__qfor all integersa,qwith ≤q<q, thus|aq| ≥q= [N–δ]. However, from (.) we have |aq| qqPNδ, this is a contradiction. We have thus established that for at least onei,P<qiQ. Hence Lemma . gives the desired inequality forW(α).

Lemma . We have

D



i=

–

α

K

(α)dαX 

–δ+ε_.

Proof By Lemmas ., . and Hölder’s inequality, we have

D



i=

Fi(λiα)G(–α)K (α)dα max

α∈D

W(α)

+∞

–∞

F(λα) 

  +∞

–∞

F(λα) 

 

+ +∞

–∞

F(λα) 

  +∞

–∞

F(λα) 

 

· 

j= +∞

–∞

Fj(λjα)



K (α)dα

 

k= +∞

–∞

Fk(λkα)



K (α)dα

 

· +∞ –∞

G(–α)K (α)dα

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X–δ+ε 

_X_+_ε _X+_ε_X_+_ε₍_NL₎_

X– 

δ+ε.

5 The trivial region

Lemma .(Lemma  of []) Let V(α) =e(αf(x, . . . ,xm)),where f is any real function and the summation is over any ﬁnite set of values of x, . . . ,xm.Then,for any A> ,we have

|α|>A

V(α)Kν(α)dα≤

A

∞

–∞

V(α)Kν(α)dα.

Lemma . We have

c



i=

–

α

K

(α)dαX 

–δ+ε_.

Proof By Lemmas ., . and Schwarz’s inequality, we have

c



i=

–

α

K (α)dα

c



i=

Fi(λiα)G(–α)

K

(α)dα 

P

+∞

–∞ 

i=

Fi(λiα)G(–α)

K

(α)dα N–δF(λα)

 

+∞

–∞

F(λα) 

 +∞

–∞

F(λα) 



· 

j= +∞

–∞

Fj(λjα)



K (α)dα

 

k= +∞

–∞

Fk(λkα)



K (α)dα

 

· +∞ –∞

G(–α)K (α)dα

 

N–δX 

_X+ε 

_X+ε 

_X_+_ε₍_NL₎

X–δ+ε_.

6 The proof of Theorem 1.1

From Lemmas ., . and . we conclude thatJ(C) X. From Lemma . it follows thatJ(D) =o(X_{). From Lemma . we have}_J_{(c) =}_o₍_X_{). Thus}

J X_, N₍_X₎ _X_L–_,

namely, under conditions of Theorem .,

λx+λx+λx+λx+λx+· · ·+λx–p–   <

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has inﬁnitely many solutions in positive integers x,x, . . . ,x and primep. It is evident from (.) that

p<λx+λx+λx+λx+λx+· · ·+λx<p+ ,

and hence

λx+λx+λx+λx+λx+· · ·+λx

=p.

The proof of Theorem . is complete.

Competing interests

The author declares that he has no competing interests.

Received: 30 August 2015 Accepted: 28 October 2015 References

1. Brüdern, J, Kawada, K, Wooley, TD: Additive representation in thin sequences, VIII: Diophantine inequalities in review. In: Number Theory: Dreaming in Dreams. Series on Number Theory and Its Applications, vol. 6, pp. 20-79 (2010) 2. Vaughan, R: The Hardy-Littlewood Method, 2nd edn. Cambridge Tracts in Mathematics, vol. 125. Cambridge University

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