# Equidistribution Modulo 1

Authors: Wiem Gadri
Journal: Journal of Mathematics
 (2021)

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## Abstract

The generalisation of questions of the classic arithmetic has long been of interest. One line of questioning, introduced by Car in 1995, inspired by the equidistribution of the sequencenαn∈N where 0<α<1, is the study of the sequence K1/l, where K is a polynomial having an l-th root in the field of formal power series. In this paper, we consider the sequence L′1/l, where L′ is a polynomial having an l-th root in the field of formal power series and satisfying L′≡BmodC. Our main result is to prove the uniform distribution in the Laurent series case. Particularly, we prove the case for irreducible polynomials.


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## Body

## 1. Introduction

In 1952, Carlitz [1] introduced the definition of equidistribution modulo 1 in the formal power series case which reveals profitable; it uses Weyl’s criterion [1], the generalisation of van der Corput inequality by Dijksma [2], and the theorem of Koksma by Mathan [3].Car in [4], inspired by equidistribution modulo 1 of the sequence nαn∈ℕ where 0<α<1, characterised equidistribution modulo 1 of the sequences L1/l and P1/l, where L describes the sequence of polynomials in FqX (resp. P describes the sequence of irreducible polynomials in FqX) with an l-th root L1/l (resp. P1/l) in the field of formal power series.In 2013, Mauduit and Car studied in [5] the Q−automaticity of the set of k-th power of polynomials in FqX. Moreover, they calculated the number of polynomials K∈FqX with degree N such that the sum of digits of Kk in base Q is fixed. In the same subject, Madritsch and Thuswaldner in [6] called the maps f:FqX⟶G, where G is the group of Q-additives satisfying fAQ+B=fA+fB for all polynomials A,B∈FqX with degB<degQ. They proved the equidistribution of the sequence hWi, where h∈FqX−1Y is a polynomial with coefficients in the field of formal power series and Wi is an ordered sequence of polynomials in CJ=A∈Ln:fA≡JmodM if and only if one of the coefficients of hY−h0 is irrational.In this article, we are interested in the subsequencesLn′ of Ln and Pn′ of Pn of polynomials in arithmetic progression having an l-th root. We will prove that the sequences Ln′1/l and Pn′1/l are equidistributed modulo 1.


## 2. Preliminary

LetFq be a finite field of characteristic p with q elements. We consider FqX, FqX, and FqX−1 as analogues of ℤ,ℚ, and ℝ, respectively.An elementf∈FqX−1 is of the form f=∑i=n0+∞aiX−i,withai∈Fq,n0∈ℤ,andan0≠0. We define νf=degf=−n0 and f=qdegf. We note f the polynomial part of f and f its fractionary part. Let Resf=a1 if f≠0, and sgnf=an0. Let ψ:Fq⟶ℂ be a nontrivial additive character. For all f∈FqX−1, we suppose that Ef=ψResf.Letl be a positive integer >2 which is not divisible by the characteristic p of the field Fq. We introduce ℒ=a1,…,ar as the set of the r-th elements having an l-th root in Fq∗, and we have(1)r=q−1l,q−1.Then, forf and g∈Fq∗X−1, g is called an l-th root of f; we note f=gl if and only if νf≡0modl and sgnf∈ℒ. In particular, a nonzero polynomial A has an l-th root in Fq∗X−1 if and only if degA≡0modl and sgnA∈ℒ.We denote byL the set of polynomials with an l-th root in Fq∗X−1:(2)L=A∈FqX∖0:degA≡0modlandsgnA∈ℒ,and if I is the set of irreducible polynomials over FqX, we define ℙ=L∩I.Ifn=∑i=1Sniqi, where ni∈0,…,q−1 for all i∈0,…,s, is the representation in base q of the integer n≥1, then let(3)Hn=χn0+⋯+χnsXs,where χni are given by the bijection ni↦χni from 0,…,q−1 to Fq. For n=0 and 1, it is convenient to suppose that χ0=0 and χ1=1. We define the order in Fq by χni<χni+1, for all ni∈0,…,q−1, and in Fq by(4)m<n⟹degHm≤degHn.Then, we orderFq by posing for all natural numbers n:(5)Hn<Hn+1.This paper is devoted to the study of equidistribution modulo 1 of a certain sequence in the field of Laurent formal power series. In 1952, Carlitz introduced and characterised equidistribution modulo 1 in the field of Laurent formal power series and obtained the following result.Lemma 1.
(see [1]; Weyl’s criterion). The sequence Θ=θn with values in FqX−1 is equidistributed modulo 1 if and only if(6)limN⟶∞1N∑n=1NEHθn=0,for all H∈FqX∗.Finally, we enounce a result which concerns a class of irreducible polynomials given by Artin in [7], which will be very useful later.Theorem 1.
(see [7]). Let C,B∈FqX be coprime polynomials. If πm:C,B denotes the number of monic irreducible polynomials with degree m which are congruent to B modulo C, then(7)πm:C,B=1ΦCqmm+Oqmθm,where θ is a constant <1. This theorem is analogous to the theorem of prime numbers in arithmetic progression.


## 3. Results

Letl≥2 be an integer nondivisible by the characteristic p of the field Fq; we order the set of the l-th powers of ℒ under the increasing order of Fq, and we fix a polynomial C with degree c. For all B∈FqX, we denote by L′ the subset of L defined in (2),(8)L′=LC,B=A∈L:A≡BmodC,and ℙ′ the subset of ℙ=L∩I:(9)ℙ′=ℙC,B=A∈L∩I:A≡BmodCwithB,C=1.We ordered the elements ofL′ and ℙ′ with the order relation defined in (2); hence,(10)L′=L1′,…,Ln′,…andℙ′=P1′,…,Pn′,….The aim of this paper is to prove the following theorems.Theorem 2.
LetLn′ be the sequence of polynomials of L′ indexed under the increasing order of FqX. Then, for l≥2, the sequence Ln′1/l is equidistributed modulo 1.Theorem 3.
LetPn′ be the sequence of polynomials of ℙ′ indexed under the increasing order of FqX. Then, the sequence Pn′1/l is equidistributed modulo 1 for l>1/1−θ, and θ is a constant defined in Theorem 1. In particular, if Xq−X does not divide C, then let l≥3.


## 4. Proofs of Theorems2 and 3

### 4.1. Tools

A generalisation of Theorem1 was proved in 1965 by Hayes introducing the arithmetic progression.Lemma 2.
(see [8]). Let C∈FqX be a polynomial with degree c. Then, for all polynomials B, there exist exactly qm−c monic polynomials with degree m which are congruent to B modulo C if m≥c.Theorem 4.
(see [8]). Let k≥1 be a positive integer, u=u1,…,uk be a sequence of k elements in Fq, and C,B∈FqX be coprime polynomials. If, for m≥k, πm;u,C,B is the number of irreducible and monic polynomials P with degree m which are congruent to B modulo C such that degP−Xm−u1Xm−1−⋯−ukXm−k<m−k, then(11)πm:u,C,B=qm−kmΦC+Oqmθm,where θ is a constant <1.Remark 1.
In particular, ifXq−X does not divide C, then (11) is verified for θ=1/2.The proofs of Theorems2 and 3 are based on Corollary 1 whose proof needs the following lemmas.Lemma 3.
(see [9], Lemma II.1.1). Let k∈ℕ, H∈FqX∗ with degree h, and A∈L with degree lk. Then, for all Z∈FqX such that degZ=z<l−1k−h−1, we have(1)
A+Z∈L(2)
ResHA1/l=ResHA+Z1/lLemma 4.
(see [9], Lemma II.1.2). Let H∈FqX∗ with degree h and k∈ℕ such that l−1k≥h. Then, for all a∈ℒ, all ξ∈Fq, and all Y∈FqX with degree <k+h, there exists unique η=ηa,Y∈Fq such that, for all Z∈FqX with degree <l−1k−h−1, we obtain(12)ξ=ResHaXlk+YXl−1k−h+ηXl−1k−h−1+Z1/l.Corollary 1.
LetH∈FqX∗ with degree h and k∈ℕ such that l−1k≥h. For all a∈ℒ, we have(i)
∑A∈AψResHA1/l=0, where A=A∈L′,degA=lk,andsgnA=a(ii)
∑A∈JψResHA1/l=Oqlkθ+k+h+1/lk, where J=A∈ℙ′,degA=lk,andsgnA=a, where θ is a constant <1Proof.
ForA=A or J, we note(13)σA:k,a=∑A∈AψResHA1/l=∑ξ∈Fqψξπξ,where πξ is the number of polynomials A∈A such that ResHA1/l=ξ, but with Lemma 4, for Y∈FqX with degree <k+h, there exists η∈Fq such that the polynomial(14)K=Ka,Y,ξ=aXlk+YXl−1k−h+ηXl−1k−h,satisfying(15)ResHK1/l=ξ.
LetZ=A−K; we denote by πlk:Y,ξ the number of polynomials A∈A such that(16)degZ<l−1k−h−1.
We obtain(17)σA:k,a=∑ξ∈Fqψξ∑Y∈FqX\\degY<k+hπlk:Y,ξ.(i)
IfA=A, then by Lemma 2, we have πlk:Y,ξ=ql−1k−h−c−1−1. With the orthogonality criterion of ψ, it results in(18)σL′:k,a=qk+hql−1k−h−c−1−1∑ξ∈Fqψξ=0.(ii)
IfA=J, then by Theorem 4, we have(19)πlk:Y,ξ=ql−1k−h−1lkΦC+Oqlkθlk,θ<1.
We deduce that(20)σℙ′:k,a=qlk−1lkΦC∑ξ∈Fqψξ+∑ξ∈FqOqlkθ+k+hlk.
Finally, with the orthogonality criterion, we obtain(21)σℙ′:k,a=Oqlkθ+k+h+1lk,withθ<1.


### 4.2. Proof of Theorem2

InFqX, there are qm−c monic polynomials which are congruent to B modulo C with degree m, and let c=degC. We denote by am (resp. bm) the number of polynomials in L′ with degree lm (resp. ≤lm). It is sufficient to verify that(22)am=rqlm−c,bm=a1+⋯+am=rqlm+1−c−ql−cql−1,where r is defined in (1). Let H∈FqX∗ with degree h, and N is an integer such that(23)N>b1+h+1/l−1,where x defines the least integer ≥x. The sequence bm is strictly increasing, and there exists a unique integer t such that(24)bt−1≤N<bt.Moreover, there exists a unique integers∈0,…,r−1, such that(25)bt−1+1+sqlt≤N<bt−1+s+1qlt.Let(26)WN=∑n=1NEHLn′1/l.To prove Theorem2, we have to show that(27)limN⟶∞1NWN=0.Using relations (24) and (25), we rewrite the sum WN to obtain(28)WN=W1+W2+W3,with(29)W1=∑n=1bt−1EHLn′1/l,W2=∑j=0s−1∑n=bt−1+1+jqltbt−1+j+1qltEHLn′1/l,W3=∑n=bt−1+1+sqltNEHLn′1/l.We start by giving an estimation of the sumW1 which concerns the polynomials of L′ with degree ≤lt−1. We have(30)W1=∑k≤t−1∑L′∈L′degL′=lksgnL′∈ℒEHLn′1/l,=∑k≤t−1l−1k<h+c∑L′∈L′degL′=lksgnL′∈ℒEHLn′1/l+∑k≤t−1l−1k≥h+c∑L′∈L′textdegL′=lktextsgnL′∈ℒEHLn′1/l.We have to just major the first part of the sum by the number of polynomials with degree<lh+c/l−1, and we apply Corollary 1 on the second part to obtain(31)W1≤qlh+c/l−1.We apply the same Corollary1 on W2 that represents the sum of polynomials with degree lt and and with signature a1,…,as−1, then(32)W2=∑i=1s−1∑L′∈L′textdegL′=lttextsgnL′∈aiEHLn′1/l=0.The polynomials inW3 can be written in the form(33)Li′=asXlt+Hni,j≤i≤N,where j=1+bt−1+sat and the sequence Hnij≤i≤N is strictly increasing in FqX.By the order relation onFqX (4), if(34)nN=c0+c1q+⋯+cmqmand is the presentation in base q of the integer nN, we have(35)HnN=χc0+χc1X+⋯+χcmXm.To estimateW3, we will distinguish two cases: when the degree of HnN is up to the integer l−1t−h−c−1 and when it is not:1st case:m≤l−1t−h−c−1. Using (34) and the fact that the sequence nij≤i≤N is strictly increasing, we obtain(36)N−j≤nN−nj≤nN≤qm+1−1.Thus,(37)W3≤N−j+1≤ql−1t−h−c.2nd case:m>l−1t−h−c−1. The polynomials Hni defined in (33) are of the form(38)Hni=y0+y1X+⋯+ymXm≤HnN=χc0+χc1X+⋯+χcmXm.LetY be the set of polynomials of the form(39)Y=yl−1t−h−c+yl−1t−h−c+1X+⋯+ymXm−l−1t+h+csuch that, for every polynomial Z with degree <l−1t−h−c, we have(40)YXl−1t−h−c+Z≤HnNIfk is the greatest index i∈j,…,N for which Li′ are written in the form(41)Li′=asXlt+YXl−1t−h−c+Z,with Y∈Y and Z being a polynomial with degree <l−1t−h−c, then we rewrite the sum W3:(42)W3=W4+W5,with(43)W4=∑i=jkEHLi′1/landW5=∑i=k+1NEHLi′1/l.We have(44)W4=∑ξ∈Fqψξπξ,where πξ is the number of couples Y,Z such that Y∈Y, Z∈FqX with degree <l−1t−h−c, and ResHasXlt+YXl−1t−h−c+Z1/l=ξ. Moreover, we have(45)πξ=∑Y∈Yπlt:Y,ξ,where πlt:Y,ξ denotes the number of polynomials L′∈L such that(46)degL′−K<l−1k−h−c−1,with(47)K=Ka,Y,ξ=aXlk+YXl−1k−h−c−1+ηXl−1k−h−c−1,which gives the same arguments presented in the proof of Corollary 1, and then we deduce that(48)W4=0.In (34), let v be the least index >l−1t−h−c such that cv≠0; then, we have(49)nN=cmqm+⋯+cvqv+cl−1t−h−c−1ql−1t−h−c−1+⋯+c0.Since all polynomialL′ of the form(50)L′=asXlt+ymXm+⋯+yvXv+yv−1Xv−1+⋯+y0,which coefficients satisfy the condition:(51)ym≤χcm,ym−1≤χcm−1,…,yv−1≤χcv−1,yv≤χcv,is less then LnN′, we obtain(52)nk≥cmqm+⋯+cv−1qv−1+q−1qv−1+⋯+q−1,nk≥nN−ql−1t−h−c,which leads to(53)W5≤N−k≤nN−nk≤ql−1t−h−c.Then, with (48) and (53), it results in(54)W3≤ql−1t−h−c.With (31), (32), and (54), we obtain(55)1N∑n=1NEHLn′1/l≤1Nqlh+c/l−1+ql−1t−h−c,and finally, with (24), we obtain(56)1N∑n=1NEHLn′1/l≤qlh+c/l−1+ql−1t−h−cql−1rqlt+1−ql−c≪q−t≪N−1/l,which ends the proof.


### 4.3. Proof of Theorem3

The proof of Theorem3 is treated as the proof of Theorem 2, and we will keep the same notations with the appropriate modifications. Let πm,C,B be the number of monic irreducible polynomials with degree m in FqX, congruent to B modulo C, satisfying with [4] the following property:(57)qmq−1−1−2qm/2≤mπm:C,B≤qmq−1.Inℙ′, we have am=rπlm:C,B, and there exist constants c1>0 and c2>0 such that(58)c1qlmm≤bm≤c2qlmm.LetH be a nonzero polynomial with degree h and N be an integer satisfying (23). We suppose now that(59)W′N=∑n=1NEHPn′1/l.With relations (24) and (25), we obtain(60)W′N=W1′+W2′+W3′.By the same method used in the proof of Theorem2, with Corollary 1, we obtain(61)W1′=∑n=1bt−1EHPn′1/l≤qlh+c/l−1+Oqt−1lθ+1+h+1lh+c,(62)W2′=Oqltθ+t+h+1lt,and then(63)W2′=∑j=0s−1∑n=bt−1+1+jqltbt−1+j+1qltEHPn′1/l=∑i=1r∑P′∈ℙ′textdegP′=lttextsgnP′=aiEHP′1/l.W4′=∑i=jkEHPi′1/l=Oqltθ+t+hlt,where W4′ is the sum defined in (42) concerning the polynomials in ℙ′. Finally, we treat the sum(64)W5′=∑i=k+1NEHPi′1/l.With Theorem4, for all polynomials(65)Y=asXlt+ymXm+⋯+yvXv+⋯+yl−1t−h−cXl−1t−h−c,in which coefficients satisfy condition (51), there exists(66)πlt:Y,C,B=ql−1t−h−cltΦC+Oqltθlt,θ<1.Irreducible polynomialsP′ are congruent to B modulo C such that degP′−Y<l−1t−h−c. Such polynomials P′ are in ℙ′, and we have(67)nk≥cmqm+⋯+cv−1qv−1+⋯+q−1ql−1t−h−cnN−2ql−1t−h−c+1.Then,(68)W5′≤N−k≤2ql−1t−h−c−1.With (63and68, it results in(69)W3′=W4′+W5′≤2ql−1t−h−c+Oqltθ+t+hlt.Finally, from (61), (62), and (69), we have(70)1N∑n=1NEHPn′1/l≤qlh+c/l−1+Oqt−1lθ+1+h+1lh+c+Oqltθ+t+h+1lt+2ql−1t−h−c+Oqltθ+t+hlt.Then,(71)1N∑n=1NEHPn′1/l≪qlθ+1t+ql−1tN≪Nθ+1/l,which gives the corresponding conclusion needed for l>1/1−θ.


## 4.1. Tools

A generalisation of Theorem1 was proved in 1965 by Hayes introducing the arithmetic progression.Lemma 2.
(see [8]). Let C∈FqX be a polynomial with degree c. Then, for all polynomials B, there exist exactly qm−c monic polynomials with degree m which are congruent to B modulo C if m≥c.Theorem 4.
(see [8]). Let k≥1 be a positive integer, u=u1,…,uk be a sequence of k elements in Fq, and C,B∈FqX be coprime polynomials. If, for m≥k, πm;u,C,B is the number of irreducible and monic polynomials P with degree m which are congruent to B modulo C such that degP−Xm−u1Xm−1−⋯−ukXm−k<m−k, then(11)πm:u,C,B=qm−kmΦC+Oqmθm,where θ is a constant <1.Remark 1.
In particular, ifXq−X does not divide C, then (11) is verified for θ=1/2.The proofs of Theorems2 and 3 are based on Corollary 1 whose proof needs the following lemmas.Lemma 3.
(see [9], Lemma II.1.1). Let k∈ℕ, H∈FqX∗ with degree h, and A∈L with degree lk. Then, for all Z∈FqX such that degZ=z<l−1k−h−1, we have(1)
A+Z∈L(2)
ResHA1/l=ResHA+Z1/lLemma 4.
(see [9], Lemma II.1.2). Let H∈FqX∗ with degree h and k∈ℕ such that l−1k≥h. Then, for all a∈ℒ, all ξ∈Fq, and all Y∈FqX with degree <k+h, there exists unique η=ηa,Y∈Fq such that, for all Z∈FqX with degree <l−1k−h−1, we obtain(12)ξ=ResHaXlk+YXl−1k−h+ηXl−1k−h−1+Z1/l.Corollary 1.
LetH∈FqX∗ with degree h and k∈ℕ such that l−1k≥h. For all a∈ℒ, we have(i)
∑A∈AψResHA1/l=0, where A=A∈L′,degA=lk,andsgnA=a(ii)
∑A∈JψResHA1/l=Oqlkθ+k+h+1/lk, where J=A∈ℙ′,degA=lk,andsgnA=a, where θ is a constant <1Proof.
ForA=A or J, we note(13)σA:k,a=∑A∈AψResHA1/l=∑ξ∈Fqψξπξ,where πξ is the number of polynomials A∈A such that ResHA1/l=ξ, but with Lemma 4, for Y∈FqX with degree <k+h, there exists η∈Fq such that the polynomial(14)K=Ka,Y,ξ=aXlk+YXl−1k−h+ηXl−1k−h,satisfying(15)ResHK1/l=ξ.
LetZ=A−K; we denote by πlk:Y,ξ the number of polynomials A∈A such that(16)degZ<l−1k−h−1.
We obtain(17)σA:k,a=∑ξ∈Fqψξ∑Y∈FqX\\degY<k+hπlk:Y,ξ.(i)
IfA=A, then by Lemma 2, we have πlk:Y,ξ=ql−1k−h−c−1−1. With the orthogonality criterion of ψ, it results in(18)σL′:k,a=qk+hql−1k−h−c−1−1∑ξ∈Fqψξ=0.(ii)
IfA=J, then by Theorem 4, we have(19)πlk:Y,ξ=ql−1k−h−1lkΦC+Oqlkθlk,θ<1.
We deduce that(20)σℙ′:k,a=qlk−1lkΦC∑ξ∈Fqψξ+∑ξ∈FqOqlkθ+k+hlk.
Finally, with the orthogonality criterion, we obtain(21)σℙ′:k,a=Oqlkθ+k+h+1lk,withθ<1.


## 4.2. Proof of Theorem2

InFqX, there are qm−c monic polynomials which are congruent to B modulo C with degree m, and let c=degC. We denote by am (resp. bm) the number of polynomials in L′ with degree lm (resp. ≤lm). It is sufficient to verify that(22)am=rqlm−c,bm=a1+⋯+am=rqlm+1−c−ql−cql−1,where r is defined in (1). Let H∈FqX∗ with degree h, and N is an integer such that(23)N>b1+h+1/l−1,where x defines the least integer ≥x. The sequence bm is strictly increasing, and there exists a unique integer t such that(24)bt−1≤N<bt.Moreover, there exists a unique integers∈0,…,r−1, such that(25)bt−1+1+sqlt≤N<bt−1+s+1qlt.Let(26)WN=∑n=1NEHLn′1/l.To prove Theorem2, we have to show that(27)limN⟶∞1NWN=0.Using relations (24) and (25), we rewrite the sum WN to obtain(28)WN=W1+W2+W3,with(29)W1=∑n=1bt−1EHLn′1/l,W2=∑j=0s−1∑n=bt−1+1+jqltbt−1+j+1qltEHLn′1/l,W3=∑n=bt−1+1+sqltNEHLn′1/l.We start by giving an estimation of the sumW1 which concerns the polynomials of L′ with degree ≤lt−1. We have(30)W1=∑k≤t−1∑L′∈L′degL′=lksgnL′∈ℒEHLn′1/l,=∑k≤t−1l−1k<h+c∑L′∈L′degL′=lksgnL′∈ℒEHLn′1/l+∑k≤t−1l−1k≥h+c∑L′∈L′textdegL′=lktextsgnL′∈ℒEHLn′1/l.We have to just major the first part of the sum by the number of polynomials with degree<lh+c/l−1, and we apply Corollary 1 on the second part to obtain(31)W1≤qlh+c/l−1.We apply the same Corollary1 on W2 that represents the sum of polynomials with degree lt and and with signature a1,…,as−1, then(32)W2=∑i=1s−1∑L′∈L′textdegL′=lttextsgnL′∈aiEHLn′1/l=0.The polynomials inW3 can be written in the form(33)Li′=asXlt+Hni,j≤i≤N,where j=1+bt−1+sat and the sequence Hnij≤i≤N is strictly increasing in FqX.By the order relation onFqX (4), if(34)nN=c0+c1q+⋯+cmqmand is the presentation in base q of the integer nN, we have(35)HnN=χc0+χc1X+⋯+χcmXm.To estimateW3, we will distinguish two cases: when the degree of HnN is up to the integer l−1t−h−c−1 and when it is not:1st case:m≤l−1t−h−c−1. Using (34) and the fact that the sequence nij≤i≤N is strictly increasing, we obtain(36)N−j≤nN−nj≤nN≤qm+1−1.Thus,(37)W3≤N−j+1≤ql−1t−h−c.2nd case:m>l−1t−h−c−1. The polynomials Hni defined in (33) are of the form(38)Hni=y0+y1X+⋯+ymXm≤HnN=χc0+χc1X+⋯+χcmXm.LetY be the set of polynomials of the form(39)Y=yl−1t−h−c+yl−1t−h−c+1X+⋯+ymXm−l−1t+h+csuch that, for every polynomial Z with degree <l−1t−h−c, we have(40)YXl−1t−h−c+Z≤HnNIfk is the greatest index i∈j,…,N for which Li′ are written in the form(41)Li′=asXlt+YXl−1t−h−c+Z,with Y∈Y and Z being a polynomial with degree <l−1t−h−c, then we rewrite the sum W3:(42)W3=W4+W5,with(43)W4=∑i=jkEHLi′1/landW5=∑i=k+1NEHLi′1/l.We have(44)W4=∑ξ∈Fqψξπξ,where πξ is the number of couples Y,Z such that Y∈Y, Z∈FqX with degree <l−1t−h−c, and ResHasXlt+YXl−1t−h−c+Z1/l=ξ. Moreover, we have(45)πξ=∑Y∈Yπlt:Y,ξ,where πlt:Y,ξ denotes the number of polynomials L′∈L such that(46)degL′−K<l−1k−h−c−1,with(47)K=Ka,Y,ξ=aXlk+YXl−1k−h−c−1+ηXl−1k−h−c−1,which gives the same arguments presented in the proof of Corollary 1, and then we deduce that(48)W4=0.In (34), let v be the least index >l−1t−h−c such that cv≠0; then, we have(49)nN=cmqm+⋯+cvqv+cl−1t−h−c−1ql−1t−h−c−1+⋯+c0.Since all polynomialL′ of the form(50)L′=asXlt+ymXm+⋯+yvXv+yv−1Xv−1+⋯+y0,which coefficients satisfy the condition:(51)ym≤χcm,ym−1≤χcm−1,…,yv−1≤χcv−1,yv≤χcv,is less then LnN′, we obtain(52)nk≥cmqm+⋯+cv−1qv−1+q−1qv−1+⋯+q−1,nk≥nN−ql−1t−h−c,which leads to(53)W5≤N−k≤nN−nk≤ql−1t−h−c.Then, with (48) and (53), it results in(54)W3≤ql−1t−h−c.With (31), (32), and (54), we obtain(55)1N∑n=1NEHLn′1/l≤1Nqlh+c/l−1+ql−1t−h−c,and finally, with (24), we obtain(56)1N∑n=1NEHLn′1/l≤qlh+c/l−1+ql−1t−h−cql−1rqlt+1−ql−c≪q−t≪N−1/l,which ends the proof.


## 4.3. Proof of Theorem3

The proof of Theorem3 is treated as the proof of Theorem 2, and we will keep the same notations with the appropriate modifications. Let πm,C,B be the number of monic irreducible polynomials with degree m in FqX, congruent to B modulo C, satisfying with [4] the following property:(57)qmq−1−1−2qm/2≤mπm:C,B≤qmq−1.Inℙ′, we have am=rπlm:C,B, and there exist constants c1>0 and c2>0 such that(58)c1qlmm≤bm≤c2qlmm.LetH be a nonzero polynomial with degree h and N be an integer satisfying (23). We suppose now that(59)W′N=∑n=1NEHPn′1/l.With relations (24) and (25), we obtain(60)W′N=W1′+W2′+W3′.By the same method used in the proof of Theorem2, with Corollary 1, we obtain(61)W1′=∑n=1bt−1EHPn′1/l≤qlh+c/l−1+Oqt−1lθ+1+h+1lh+c,(62)W2′=Oqltθ+t+h+1lt,and then(63)W2′=∑j=0s−1∑n=bt−1+1+jqltbt−1+j+1qltEHPn′1/l=∑i=1r∑P′∈ℙ′textdegP′=lttextsgnP′=aiEHP′1/l.W4′=∑i=jkEHPi′1/l=Oqltθ+t+hlt,where W4′ is the sum defined in (42) concerning the polynomials in ℙ′. Finally, we treat the sum(64)W5′=∑i=k+1NEHPi′1/l.With Theorem4, for all polynomials(65)Y=asXlt+ymXm+⋯+yvXv+⋯+yl−1t−h−cXl−1t−h−c,in which coefficients satisfy condition (51), there exists(66)πlt:Y,C,B=ql−1t−h−cltΦC+Oqltθlt,θ<1.Irreducible polynomialsP′ are congruent to B modulo C such that degP′−Y<l−1t−h−c. Such polynomials P′ are in ℙ′, and we have(67)nk≥cmqm+⋯+cv−1qv−1+⋯+q−1ql−1t−h−cnN−2ql−1t−h−c+1.Then,(68)W5′≤N−k≤2ql−1t−h−c−1.With (63and68, it results in(69)W3′=W4′+W5′≤2ql−1t−h−c+Oqltθ+t+hlt.Finally, from (61), (62), and (69), we have(70)1N∑n=1NEHPn′1/l≤qlh+c/l−1+Oqt−1lθ+1+h+1lh+c+Oqltθ+t+h+1lt+2ql−1t−h−c+Oqltθ+t+hlt.Then,(71)1N∑n=1NEHPn′1/l≪qlθ+1t+ql−1tN≪Nθ+1/l,which gives the corresponding conclusion needed for l>1/1−θ.



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*Source: 1357859-2021-02-19.xml*