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Research article

Fueling economies through credit and industrial activities. A way of financing sustainable economic development in Brazil

  • As an emerging country preoccupied with preserving its resources for the future, Brazil aims to find the right balance between economic advancement and sustainability goals. In this context, we tackled the link between sustainable economic development and macroeconomic indicators related to domestic credit granted to the private sector, CO2 emissions from industries, and annual inflation rate. By means of time series data analysis run for the period 1996‒2022 via three estimation methods (i.e., least squares, fully modified least squares, dynamic least squares), we found that annual GDP growth rate, control of corruption, and rule of law (as proxies for sustainable economic development) are significantly impacted by GDP growth rate and emissions. Therefore, access to financial resources and intensive industrial activities yielding emissions trigger economic growth, tend to strengthen control of corruption and the rule of law. Additionally, policy implications and future research directions are addressed.

    Citation: Larissa M. Batrancea, Anca Nichita, Horia Tulai, Mircea-Iosif Rus, Ema Speranta Masca. Fueling economies through credit and industrial activities. A way of financing sustainable economic development in Brazil[J]. Green Finance, 2025, 7(1): 24-39. doi: 10.3934/GF.20250002

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  • As an emerging country preoccupied with preserving its resources for the future, Brazil aims to find the right balance between economic advancement and sustainability goals. In this context, we tackled the link between sustainable economic development and macroeconomic indicators related to domestic credit granted to the private sector, CO2 emissions from industries, and annual inflation rate. By means of time series data analysis run for the period 1996‒2022 via three estimation methods (i.e., least squares, fully modified least squares, dynamic least squares), we found that annual GDP growth rate, control of corruption, and rule of law (as proxies for sustainable economic development) are significantly impacted by GDP growth rate and emissions. Therefore, access to financial resources and intensive industrial activities yielding emissions trigger economic growth, tend to strengthen control of corruption and the rule of law. Additionally, policy implications and future research directions are addressed.



    Let q be an integer. For each integer a with

    1a<q,  (a,q)=1,

    we know that [1] there exists one and only one ˉa with

    1ˉa<q

    such that

    aˉa1(q).

    Define

    R(q):={a:1aq,(a,q)=1,2a+ˉa},
    r(q):=#R(q).

    The work [2] posed the problem of investigating a nontrivial estimation for r(q) when q is an odd prime. Zhang [3,4] gave several asymptotic formulas for r(q), one of which is:

    r(q)=12ϕ(q)+O(q12d2(q)log2q),

    where ϕ(q) is the Euler function and d(q) is the divisor function. Lu and Yi [5] studied a generalization of the Lehmer problem over short intervals. Let n2 be a fixed positive integer, q3 and c be integers with

    (nc,q)=1.

    They defined

    rn(θ1,θ2,c;q)=#{(a,b)[1,θ1q]×[1,θ2q]abc( mod q),na+b},

    where 0<θ1,θ21, and obtained

    rn(θ1,θ2,c;q)=(11n)θ1θ2φ(q)+O(q1/2τ6(q)log2q),

    where the O constant depends only on n. In addition, Xi and Yi [6] considered generalized Lehmer problem over short intervals. Han and Liu [7] gave an upper bound estimation for another generalization of the Lehmer problem over incomplete interval.

    Guo and Yi [8] also found the Lehmer problem has good distribution properties on Beatty sequences. For fixed real numbers α and β, defined by

    Bα,β:=(αn+β)n=1.

    Beatty sequences are linear sequences. Based on the results obtained, we conjecture the Lehmer problem also has good distribution properties in some non-linear sequences.

    The Piatetski-Shapiro sequence is a non-linear sequence, defined by

    Nc={nc:nN},

    where cR is non-integer with c>1 and zR. This sequence was first introduced by Piatetski-Shapiro [9] to study prime numbers in sequences of the form f(n), where f(n) is a polynomial. A positive integer is called square-free if it is a product of distinct primes. The distribution of square-free numbers in the Piatetski-Shapiro sequence has been studied extensively. Stux [10] found that, as x tends to infinity,

    nxnc is square-free 1=(6π2+o(1))x, for  1<c<43. (1.1)

    In 1978, Rieger [11] improved the range to 1<c<3/2 and obtained

    6xπ2+O(x(2c+1)/4+ε), for  1<c<32

    Considering the results obtained, we develop this problem by investigating

    R(c;q):=nNcR(q)n is square-free 1

    and range of c when q tends to infinity. By methods of exponential sum and Kloosterman sums and fairly detailed calculations, we get the following result, which is significant for understanding the distribution properties of the Lehmer problem.

    Theorem 1.1. Let q be an odd integer and large enough,

    γ:=1/candc(1,43),

    we obtain

    R(c;q)=3π2pq(1+p1)1qγ+O(pq(1p12)1qγ12)+O(q713γ+413pq(1p12)1logq)+O(q34d3(q)logq)+O(qγ16d2(q)log3q),

    where the O constant only depends on c.

    This paper consists of three main sections. Introduction covers the origins and developments of the Lehmer problem, along with several interesting results. It also presents relevant findings related to the Piatetski-Shapiro sequences. The second section includes some definitions and lemmas throughout the paper. The third section outlines the calculation process, where we use additive characteristics to convert the congruence equations into exponential sum problems. We then employ the Kloosterman sums and exponential sums methods to derive an interesting asymptotic formula.

    To complete the proof of the theorem, we need the following several definitions and lemmas.

    In this paper, we denote by t and {t} the integral part and the fractional part of t, respectively. As is customary, we put

    e(t):=e2πitand{t}:=tt.

    The notation t is used to denote the distance from the real number t to the nearest integer; that is,

    t:=minnZ|tn|.

    And indicates that the variable summed over takes values coprime to the number q. Throughout the paper, ε always denotes an arbitrarily small positive constant, which may not be the same at different occurrences; the implied constants in symbols O,, and may depend (where obvious) on the parameters c and ε, but are absolute otherwise. For given functions F and G, the notations

    FG,   GF   andF=O(G)

    are all equivalent to the statement that the inequality

    |F|C|G|

    holds with some constant C>0.

    Lemma 2.1. Let 1c(m) denote the characteristic function of numbers in a Piatetski-Shapiro sequence, then

    1c(m)=γmγ1+O(mγ2)+ψ((m+1)γ)ψ(mγ),

    where

    ψ(t)=tt12   andγ=1/c.

    Proof. Note that an integer m has the form

    m=nc

    for some integer n if and only if

    mnc<m+1,(m+1)γ<nmγ.

    So

    1c(m)=mγ(m+1)γ=mγψ(mγ)+(m+1)γ+ψ((m+1)γ)=γmγ1+O(mγ2)+ψ((m+1)γ)ψ(mγ).

    Thie completes the proof.

    Lemma 2.2. Let H1 be an integer, ah,bh be real numbers, we have

    |ψ(t)0<|h|Hahe(th)||h|Hbhe(th),ah1|h|,bh1H.

    Proof. In 1985, Vaaler showed how Beurling's function could be used to construct a trigonometric polynomial approximation to ψ(x). For each positive integer N, Vaaler's construction yields a trigonometric polynomial ψ of degree N which satisfies

    |ψ(x)ψ(x)|12N+2|n|N(1|n|N+1)e(nx),

    where

    ψ(x)=1|n|N(2πin)1ˆJN+1(n)e(nx),H(z)=sin2πzπ2{n=sgn(n)(zn)2+2z},J(z)=12H(z),HN(z)=sin2πzπ2{|n|Nsgn(n)(zn)2+2z},JN(z)=12HN(z),

    and sgn(n) is the sign of n. The Fourier transform ˆJ(t) satisfies

    ˆJ(t)={1,t=0;πt(1|t|)cotπt+|t|,0<|t|<1;0,t1.

    To be short, we denote

    ah=(2πih)1ˆJH+1(h)1|h|,bh=12H+2(1|h|H+1)1H.

    There are more details in Appendix Theorem A.6. of [12].

    Lemma 2.3. Denote

    Kl(m,n;q)=qa=1qb=1ab1(modq)e(ma+nbq),

    then

    Kl(m,n;q)(m,n,q)12q12d(q),

    where (m,n,q) is the greatest common divisor of m,n and q and d(q) is the number of positive divisors of q.

    Proof. The proof is given in [13].

    Lemma 2.4. (Korobov [14]) Let α be a real number, Q be an integer, and P be a positive integer, then

    |Q+Px=Q+1e(αx)|min(P,12α).

    Lemma 2.5. (Karatsuba [15]) For any number b, U<0, K1, let

    a=sr+θr2,(r,s)=1,   r1,   |θ|1,

    then

    kKmin(U,1ak+b)(Kr+1)(U+rlogr).

    Lemma 2.6. Suppose f is continuously differentiable, f(n) is monotonic, and

    f(n)λ1>0

    on I, then

    nIe(f(n))λ11.

    Proof. See [12, Theorem 2.1].

    Lemma 2.7. Let k be a positive integer, k2. Suppose that f(n) is a real-valued function with k continuous derivatives on [N,2N], Further suppose that

    0<Ff(k)(n)hF.

    Then

    |N<x2Ne(f(n))|FκNλ+F1,

    where the implied constant is absolute.

    Proof. See [12, Chapter 3].

    By the definition of Mobius function

    μ(n)={(1)ω(n),p|n,p2n,0,p2|n,

    it is clear that n is square-free if and only if

    μ2(n)=1,

    where ω(n) is the number of prime divisor of n. So

    R(c;q)=12qn=1(1(1)n+ˉn)μ2(n)1c(n)=12(R1R2), (3.1)

    where

    R1=qn=1μ2(n)1c(n)

    and

    R2=qn=1(1)n+ˉnμ2(n)1c(n).

    From Lemma 2.1, we have

    R1=qn=1μ2(n)1c(n)=qn=1μ2(n)(γnγ1+O(nγ2)+ψ((n+1)γ)ψ(nγ))=R11+R12, (3.2)

    where

    R11:=qn=1μ2(n)(γnγ1+O(nγ2))=qn=1μ2(n)γnγ1+O(qn=1μ2(n)nγ2).

    Let

    D={d:p|dp|q}

    and λ(n) is Liouville function. When nR(q),

    μ2(n)={dm=n,dDλ(d)μ2(m),(n,q)=1,0,(n,q)>1. (3.3)

    We just consider the first term of R11. Applying Euler summuation [1],

    qn=1μ2(n)γnγ1=qn=1dm=ndDλ(d)μ2(m)γ(dm)γ1=dDλ(d)dγ1mqdμ2(m)γmγ1=dDλ(d)dγ1mqd(l2mμ(l))γmγ1=dDλ(d)dγ1l(qd)12μ(l)l2γ2mqdl2γmγ1=dDλ(d)dγ1l(qd)12μ(l)l2γ2((qdl2)γ+O((qdl2)γ1))=qγdDλ(d)d1l(qd)12μ(l)l2+O(dDl(qd)12qγ1)=qγdDλ(d)d1(lμ(l)l2+O((qd)12))+O(pq(1p12)1qγ12)=6π2pq(1+p1)1qγ+O(pq(1p12)1qγ12),

    thus

    R11=6π2pq(1+p1)1qγ+O(pq(1p12)1qγ12). (3.4)

    For R12, by Lemma 2.2, we have

    R12:=qn=1μ2(n)(ψ((n+1)γ)ψ(nγ))=R121+O(R122), (3.5)

    where

    R121:=qn=1μ2(n)(0<|h|Hah(e((n+1)γh)e(nγh)))

    and

    R122:=qn=1μ2(n)(|h|Hbh(e((n+1)γh)+e(nγh))).

    Define

    f(t)=e(((dt)γ(dt+1)γ)h)1,

    then

    f(t) |h|(dt)γ1,f(t)t|h|dγ1tγ2.

    By Lemma 2.2 and Eq (3.3),

    R121=0<|h|HahdDλ(d)(1<mqdμ2(m)e((dm)γh)f(m))0<|h|H|h|1dD|qd0f(t)d(1<mtμ2(m)e((dm)γh))|0<|h|H|h|1dD|f(qd)(1<mqdμ2(m)e((dm)γh))|+0<|h|H|h|1dD|qd0f(t)t1<mtμ2(m)e((dm)γh)dt|.

    Let

    (κ,λ)=(16,23)

    be an exponential pair. Applying Lemma 2.7, it's easy to see

    1<mtμ2(m)e((dm)γh)=mt(l2mμ(l))e((dm)γh)lt12|mtl2e((dl2m)γh)|lt12logq(((dl2)γ|h|(tl2)γ1)16(tl2)23+((dl2)γ|h|(tl2)γ1)1)logqlt12((dγ|h|)16t16γ+12l1+(dγ|h|)1t1γl2)(dγ|h|)16t16γ+12log2q+(dγ|h|)1t1γlogq(dγ|h|)16t16γ+12log2q,

    thus

    R1210<|h|H|h|1dD|h|qγ1(dγ|h|)16(qd)16γ+12log2q+0<|h|H|h|1dDqd0|h|dγ1tγ2(dγ|h|)16t16γ+12log2qdt0<|h|H|h|16dDd12q76γ12log2q+0<|h|H|h|16dDd76γ1log2qqd0t76γ32dtH76pq(1p12)1q76γ12log2q. (3.6)

    For R122, the contribution from h0 can be bounded by similar methods of Eq (3.6). Taking

    H=q913713γ1,

    we obtain

    R122=b0qn=1μ2(n)+0<|h|Hbhqn=1μ2(n)(e((n+1)γh)+e(nγh))H1q+H76pq(1p12)1q76γ12logqq713γ+413pq(1p12)1log2q. (3.7)

    It follows from Eqs (3.5)–(3.7),

    R12q713γ+413pq(1p12)1log2q.

    Hence

    R1=6π2pq(1+p1)1qγ+O(pq(1p12)1qγ12)+O(q713γ+413pq(1p12)1log2q). (3.8)

    Similarly,

    R2=qn=1(1)n+ˉnμ2(n)1c(n)=RP21+RP22, (3.9)

    where

    R21=qn=1(1)n+ˉnμ2(n)(γnγ1+O(nγ2))

    and

    R22=qn=1(1)n+ˉnμ2(n)(ψ((n+1)γ)ψ(nγ)).

    We also just consider the first term of R21.

    qn=1(1)n+ˉnμ2(n)γnγ1=qn=1(1)n+ˉn(d2nμ(d))γnγ1=qn=1(1)n+ˉnd2ndq14μ(d)γnγ1+qn=1(1)n+ˉnd2nq14<dq12μ(d)γnγ1. (3.10)

    It is easy to see

    qn=1(1)n+ˉnd2nq14<dq12μ(d)γnγ1qn=1d2nq14<dq12γnγ1qγ14. (3.11)

    Since for integers m and a, one has

    1qqs=1e(s(ma)q)={1,ma( mod q);0,ma( mod q).

    This gives

    qn=1(1)n+ˉnd2ndq14μ(d)γnγ1=qn=1qm=1nm1(modq)(1)n+md2ndq14μ(d)γnγ1q1a=1a=mq1b=1b=n1=qn=1qm=1nm1(modq)q1a=1a=mq1b=1b=n(1)a+bd2bdq14μ(d)γbγ1=qn=1qm=1nm1(modq)q1a=1am(modq)q1b=1bn(modq)(1)a+bd2bdq14μ(d)γbγ1=qn=1qm=1nm1(modq)q1a=1q1b=1(1)a+bd2bdq14μ(d)γbγ1×(1qqs=1e(s(ma)q))(1qqt=1e(t(nb)q))=1q2qs=1qt=1(nm1(modq)e(sm+tnq))×(q1a=1(1)ae(saq))(q1b=1(1)be(tbq)d2bdq14μ(d)γbγ1). (3.12)

    From Lemma 2.3,

    nm1(modq)e(sm+tnq)=Kl(s,t;q)(s,t,q)12q12d(q). (3.13)

    Note the estimate

    |q1a=1(1)ae(saq)|1|e(12sq)1|1|cossqπ| (3.14)

    holds. By Abel summation and Lemma 2.4, we have

    q1b=1(1)be(tbq)d2bdq14μ(d)γbγ1=dq14μ(d)q1d2b=1(1)d2be(td2bq)γ(d2b)γ1dq14d2(γ1)|q1d2b=1(1)d2be(td2bq)γbγ1|dq14d2(γ1)γ(qd2)γ1max1βq1d2|βb=1(1)d2be(td2bq)|dq142dγqγ1max1βq1d2|βb=1e(td2bq)|+dq142dγqγ1max1βq1d2|βb=1(1)be(td2bq)|dq142dqγ1min(q1d2,12d2qt)+dq142dqγ1min(q1d2,1212d2qt). (3.15)

    To be short, combining Eqs (3.13)–(3.15), we denote

    R211:=q2qs=1qt=1(s,t,q)12d(q)q121|cossqπ|dq142dqγ1min(q1d2,12d2qt)qγ3qs=1qt=1(s,t,q)12d(q)q121|cossqπ|dq14min(q1d2,12d2qt)qγ3uqu12d(q)q12qus=11|cossuqπ|dq14qut=1min(q1d2,12d2qut)

    and

    R212:=q2qs=1qt=1(s,t,q)12d(q)q121|cossqπ|dq142dqγ1min(q1d2,1212d2qt).

    Let

    (d2,qu)=r,   (d2r,qur)=1,

    making use of Lemma 2.5, we have

    qut=1min(q1d2,12d2qut)(ququr+1)(q1d2+qurlogq)qrd2+qulogq.

    Insert it to R211, then

    R211qγ3uqu12d(q)q12qus=11|12suq|dq14(d2,qu)=r(qrd2+qulogq)qγ3uqu12d(q)q12qulogqdq14r|d2r|qu(qrd2+qulogq)qγ3uqu12d(q)q12qulogqr|qudq14r12(qd2+qulogq)qγ3uqu12d(q)q12qulogq(qd(q)+q54ud(q)logq)qγ14d3(q)log3q.

    By the same method of R211,

    R212qγ14d3(q)log3q.

    Following from Eqs (3.10) and (3.11), estimations of R211 and R212,

    R21qγ14+RP211+RP212qγ14d3(q)log3q. (3.16)

    By the similar method of R12 and R21,

    R22=R221+O(R222), (3.17)

    where

    R221:=qn=1(1)n+ˉnμ2(n)(0<|h|Hah(e((n+1)γh)e(nγh)))

    and

    R222:=qn=1(1)n+ˉnμ2(n)(|h|Hbh(e((n+1)γh)+e(nγh))).

    It is obvious that

    R221=qn=1(1)n+ˉn(d2nμ(d))(0<|h|Hah(e((n+1)γh)e(nγh)))=qn=1(1)n+ˉnd2ndq16μ(d)(0<|h|Hah(e((n+1)γh)e(nγh)))+qn=1(1)n+ˉnd2nq16<dq12μ(d)(0<|h|Hah(e((n+1)γh)e(nγh))). (3.18)

    From the estimate

    e(nγh(n+1)γh)1(nγ(n+1)γ)hγnγ1h,

    by partial summation,

    qn=1(1)n+ˉnd2nq16<dq12μ(d)(0<|h|Hah(e((n+1)γh)e(nγh)))qn=1d2nq16<dq12|0<|h|Hahe(nγh)(e(nγh(n+1)γh)1)|qn=1d2nq16<dq12γnγ1HlogHqγ16HlogH. (3.19)

    For another term of R_{221} ,

    \begin{align} &\mathop{{\sum}^{\prime}}_{n = 1}^{q} (-1)^{n+\bar{n}} \mathop{\sum\limits_{d^{2} \mid n}}_{d \leqslant q^{\frac{1}{6}} }\mu(d) \left( \sum\limits_{0 < |h|\leqslant H}a_{h} \left(\mathbf{e}\left(-(n+1)^{\gamma}h\right)-\mathbf{e}(-n^{\gamma}h)\right)\right)\\ & = \mathop{{\sum}^{\prime}}_{n = 1}^{q} (-1)^{n+\bar{n}} \mathop{\sum\limits_{d^{2} \mid n}}_{d \leqslant q^{\frac{1}{6}} }\mu(d) \left( \sum\limits_{0 < |h|\leqslant H}a_{h} \left(\mathbf{e}\left(-(n+1)^{\gamma}h\right)-\mathbf{e}(-n^{\gamma}h)\right)\right) \\ &\quad\times \left(\frac{1}{q}\sum\limits_{a = 1}^{q}\sum\limits_{s = 1}^{q}\mathbf{e}(\frac{s(m-a)}{q}) \right)\left(\frac{1}{q}\sum\limits_{b = 1}^{q}\sum\limits_{t = 1}^{q}\mathbf{e}(\frac{t(n-b)}{q})\right) \\ & = \frac{1}{q^{2}}\sum\limits_{s = 1}^{q}\sum\limits_{t = 1}^{q} \left( \mathop{\sum}_{nm \equiv 1 (\bmod q)} \mathbf{e}(\frac{sm+tn}{q}) \right)\left(\sum\limits_{a = 1}^{q-1}(-1)^{a}\mathbf{e}(-\frac{sa}{q})\right)\\ &\quad\times \left(\sum\limits_{b = 1}^{q-1}(-1)^{b}\mathbf{e}(-\frac{tb}{q}) \mathop{\sum\limits_{d^{2} \mid b}}_{d \leqslant q^{\frac{1}{6} } }\mu(d) \sum\limits_{0 < |h|\leqslant H}a_{h} \left(\mathbf{e}\left(-(b+1)^{\gamma}h\right)-\mathbf{e}(-b^{\gamma}h)\right) \right ). \end{align} (3.20)

    We just need to give an estimation of the last part in (3.20). Similarly, let

    g (x) = \mathbf{e}\left(\left((d^{2}x)^{\gamma}-(d^{2}x+1)^{\gamma}\right)h \right)-1,

    then

    \begin{align} g (x) &\ll\ |h| (d^2x)^{\gamma-1}, \\ \frac{\partial g(x)}{\partial x} &\ll |h| d^{2\gamma-2}x^{\gamma-2}. \end{align}

    By partial summation,

    \begin{align} &\sum\limits_{b = 1}^{q-1}(-1)^{b}\mathbf{e}(-\frac{tb}{q}) \mathop{\sum\limits_{d^{2} \mid b}}_{d \leqslant q^{\frac{1}{6} } }\mu(d) \sum\limits_{0 < |h|\leqslant H}a_{h} \left(\mathbf{e}\left(-(b+1)^{\gamma}h\right)-\mathbf{e}(-b^{\gamma}h)\right) \\ & = \sum\limits_{0 < |h|\leqslant H}a_{h} \sum\limits_{d \leqslant q^{ \frac{1}{6} }}\mu(d)\sum\limits_{1 \leqslant b \leqslant \lfloor\frac{q-1}{d^2}\rfloor} \mathbf{e}\left((\frac{d^{2}}{2}-\frac{td^{2}}{q})b-(d^{2}b)^{\gamma}h\right)g(b), \\ & \ll\sum\limits_{0 < |h|\leqslant H}|h|^{-1} \sum\limits_{d \leqslant q^{ \frac{1}{6} }} \left|\int_{ 1}^{ \lfloor\frac{q-1}{d^2}\rfloor} g(x) \mathrm{d}\left(\sum\limits_{1 < b \leqslant x} \mathbf{e}\left((\frac{d^{2}}{2}-\frac{td^{2}}{q})b-(d^{2}b)^{\gamma}h\right) \right) \right|\\ & \ll\sum\limits_{0 < |h|\leqslant H}|h|^{-1} \sum\limits_{d \leqslant q^{ \frac{1}{6} }} \left|g( \lfloor\frac{q-1}{d^2}\rfloor) \sum\limits_{ 1 \leqslant b \leqslant \lfloor\frac{q-1}{d^2}\rfloor} \mathbf{e}\left((\frac{d^{2}}{2}-\frac{td^{2}}{q})b-(d^{2}b)^{\gamma}h\right)\right|\\ &\quad +\sum\limits_{0 < |h|\leqslant H}|h|^{-1} \sum\limits_{d \leqslant q^{ \frac{1}{6} }} \left|\int_{ 1}^{ \lfloor\frac{q-1}{d^2}\rfloor}\frac{\partial g(x)}{\partial x} \sum\limits_{ 1 \leqslant b \leqslant x} \mathbf{e}\left((\frac{d^{2}}{2}-\frac{td^{2}}{q})b-(d^{2}b)^{\gamma}h\right) \mathrm{d}x\right|, \end{align}

    where

    g( \lfloor\frac{q-1}{d^2}\rfloor) \ll |h|q^{\gamma-1}

    and

    \begin{align} &\sum\limits_{ 1 \leqslant b \leqslant \lfloor\frac{q-1}{d^2}\rfloor} \mathbf{e}\left((\frac{d^{2}}{2}-\frac{td^{2}}{q})b-(d^{2}b)^{\gamma}h\right) \\ & = \sum\limits_{ 1 \leqslant b \leqslant q^{\frac{1}{6}} } \mathbf{e}\left((\frac{d^{2}}{2}-\frac{td^{2}}{q})b-(d^{2}b)^{\gamma}h\right)+\sum\limits_{ q^{\frac{1}{6}} < b \leqslant \lfloor\frac{q-1}{d^2}\rfloor} \mathbf{e}\left((\frac{d^{2}}{2}-\frac{td^{2}}{q})b-(d^{2}b)^{\gamma}h\right). \end{align}

    It is obvious that

    \begin{align} \sum\limits_{ 1 \leqslant b \leqslant q^{\frac{1}{6}} } \mathbf{e}\left((\frac{d^{2}}{2}-\frac{td^{2}}{q})b-(d^{2}b)^{\gamma}h\right)\ll q^{\frac{1}{6}}. \end{align}

    Suppose q be large enough and for b > q^{\frac{1}{6}} , when 2 \nmid d or q \nmid td^2 ,

    \left\| (\frac{d^{2}}{2}-\frac{td^{2}}{q})-\gamma d^{2\gamma}b^{\gamma-1}h\right\|^{-1} \geqslant \frac{1}{2} \left\| (\frac{1}{2}-\frac{t }{q})d^{2}\right\|^{-1} > 0,

    and applying Lemma 2.6, we have

    \begin{align} \sum\limits_{q^{\frac{1}{6}} < b \leqslant \lfloor\frac{q-1}{d^2}\rfloor} \mathbf{e}\left((\frac{d^{2}}{2}-\frac{td^{2}}{q})b-(d^{2}b)^{\gamma}h\right) \ll& \left\| (\frac{d^{2}}{2}-\frac{td^{2}}{q})-\gamma d^{2\gamma}b^{\gamma-1}h\right\|^{-1} \\ \ll& \left\| (\frac{1}{2}-\frac{t }{q})d^{2}\right\|^{-1} . \end{align}

    So

    \sum\limits_{ 1 \leqslant b \leqslant \lfloor\frac{q-1}{d^2}\rfloor} \mathbf{e}\left((\frac{d^{2}}{2}-\frac{td^{2}}{q})b-(d^{2}b)^{\gamma}h\right) \ll \begin{cases} q^{\frac{1}{6}}+ \left\| (\frac{1}{2}-\frac{t }{q})d^{2}\right\|^{-1} , & 2 \nmid d \text{ or } q \nmid td^2;\\ \frac{q}{d^2}, & 2 \mid d \text{ and } q \mid td^2 ;\end{cases}

    which means

    \begin{align} &\sum\limits_{b = 1}^{q-1}(-1)^{b}\mathbf{e}(-\frac{tb}{q}) \mathop{\sum\limits_{d^{2} \mid b}}_{d \leqslant q^{\frac{1}{6} } }\mu(d) \sum\limits_{0 < |h|\leqslant H}a_{h} \left(\mathbf{e}\left(-(b+1)^{\gamma}h\right)-\mathbf{e}(-b^{\gamma}h)\right) \\ &\ll\sum\limits_{0 < |h|\leqslant H} |h|^{-1}\mathop{\sum\limits_{d \leqslant q^{ \frac{1}{6} }}}_{2 \nmid d \text{ or } q \nmid td^2} |h|q^{\gamma-1} \left(q^{\frac{1}{6}}+ \left\| (\frac{1}{2}-\frac{t }{q})d^{2}\right\|^{-1}\right) +\sum\limits_{0 < |h|\leqslant H} |h|^{-1}\mathop{\sum\limits_{d \leqslant q^{ \frac{1}{6} }}}_{2 \mid d \text{ and } q \mid td^2} |h|q^{\gamma} d^{-2} \\ &\ll H q^{ \gamma -1 }\left(q^{\frac{1}{3}}+ \mathop{\sum\limits_{d \leqslant q^{ \frac{1}{6} }}}_{2 \nmid d \text{ or } q \nmid td^2}\left\| (\frac{1}{2}-\frac{t }{q})d^{2}\right\|^{-1}\right)+ H q^{ \gamma }\mathop{\sum\limits_{d \leqslant q^{ \frac{1}{6} }}}_{2 \mid d \text{ and } q \mid td^2}d^{-2} . \end{align}

    We denote

    \begin{align} T(c)&: = \sum\limits_{d\leqslant q^{\frac{1}{6} }} \# \left\{ (\frac{q}{u}-2t)d^{2} \equiv c (\bmod 2\frac{q}{u}), t\leqslant \frac{q}{u}\right\} \\ &\ll \sum\limits_{d\leqslant q^{\frac{1}{6} }} (\frac{q}{u}, d^{2}) \\ &\ll q^{\frac{1}{3} }d(q), \end{align}

    thus,

    \begin{align} &\mathop{{\sum}^{\prime}}_{n = 1}^{q} (-1)^{n+\bar{n}} \mathop{\sum\limits_{d^{2} \mid n}}_{d \leqslant q^{\frac{1}{6}} }\mu(d) \left( \sum\limits_{0 < |h|\leqslant H}a_{h} \left(\mathbf{e}\left(-(n+1)^{\gamma}h\right)-\mathbf{e}(-n^{\gamma}h)\right)\right)\\ & \ll q^{-2}\sum\limits_{s = 1}^{q}\sum\limits_{t = 1}^{q}(s,t,q)^{\frac{1}{2}}d(q)q^{\frac{1}{2}}\frac{H q^{ \gamma- 1}}{|\cos\frac{s}{q}\pi|}\left(q^{\frac{1}{ 3}}+ \mathop{\sum\limits_{d \leqslant q^{ \frac{1}{6} }}}_{2 \nmid d \text{ or } q \nmid td^2} \left\| (\frac{1}{2}-\frac{t }{q})d^{2}\right\|^{-1}\right) \\ &\quad+q^{-2}\sum\limits_{s = 1}^{q}\sum\limits_{t = 1}^{q}(s,t,q)^{\frac{1}{2}}d(q)q^{\frac{1}{2}}\frac{H q^{ \gamma}}{|\cos\frac{s}{q}\pi|}\mathop{\sum\limits_{d \leqslant q^{ \frac{1}{6} }}}_{2 \mid d \text{ and } q \mid td^2}d^{-2} \\ & \ll Hq^{ \gamma-\frac{5}{2}}\sum\limits_{u\mid q}u^{\frac{1}{2}} d(q) \sum\limits_{s = 1}^{\frac{q}{u}}\frac{1}{|1-2 \frac{su}{q} |} \sum\limits_{t = 1}^{\frac{q}{u}} \left(q^{\frac{1}{3}}+ \sum\limits_{d \leqslant q^{\frac{1}{6} }} \left\| (\frac{1}{2}-\frac{ut }{q})d^{2}\right\|^{-1} \right) \\ &\quad+Hq^{ \gamma-\frac{3}{2}} \sum\limits_{u\mid q}u^{\frac{1}{2}} d(q) \sum\limits_{s = 1}^{\frac{q}{u}}\frac{1}{|1-2 \frac{su}{q} |} \sum\limits_{d \leqslant q^{ \frac{1}{6} }}\mathop{\sum\limits_{t = 1}^{\frac{q}{u}}}_{ q \mid td^2}d^{-2} \\ & \ll H q^{ \gamma-\frac{1}{6}} d^2(q) \log q+ Hq^{ \gamma-\frac{1}{3}}d^2(q) \log q \\ &\quad+ Hq^{ \gamma- \frac{3}{2}}\sum\limits_{u\mid q}u^{-\frac{1}{2}} d(q) \log^{2}q \max\limits_{C} \sum\limits_{C < c\leqslant2C} \|\frac{C}{2\frac{q}{u}}\|^{-1} T(c) \\ &\ll Hq^{ \gamma -\frac{1}{6}}d^2(q) \log^2 q. \end{align} (3.21)

    With Eqs (3.18) and (3.19), we have

    \begin{align} R_{221}&\ll Hq^{ \gamma -\frac{1}{6}}d^2(q) \log^2 q+H q^{ \gamma -\frac{1}{6}} \log H. \end{align} (3.22)

    For R_{222} , the contribution from h = 0 can be bounded by similar methods of R_{21} , and the contribution from h \neq 0 can be bounded by similar methods of R_{221} . Taking

    H = \log q ,

    we obtain

    \begin{align} R_{222}& = b_{0} \mathop{{\sum}^{\prime}}_{n = 1}^{q} (-1)^{n+\bar{n}} \mu^{2}(n) +\mathop{{\sum}^{\prime}}_{n = 1}^{q} (-1)^{n+\bar{n}}\mu^{2}(n)\left(\sum\limits_{0 < |h|\leqslant H}b_{h} \left(e\left(-(n+1)^{\gamma}h\right)+e\left(-n^{\gamma}h\right)\right)\right) \\ &\ll H^{-1}q^{\frac{3}{4}}d^{3}(q)\log^2 q +Hq^{ \gamma -\frac{1}{6}}d^2(q) \log^2 q\\ &\ll q^{\frac{3}{4}}d^{3}(q)\log q+ q^{ \gamma -\frac{1}{6}}d^{2}(q) \log^3 q . \end{align} (3.23)

    Following from Eqs (3.16), (3.22), and (3.23),

    \begin{align} R_{2}\ll q^{\gamma-\frac{1}{6}} d^{2}(q) \log^3 q+ q^{\frac{3}{4}}d^{3}(q)\log q . \end{align} (3.24)

    Hence, from Eqs (3.1), (3.8), and (3.24), we derive that

    \begin{align} R(c;q) = &\frac{3}{\pi^{2}}\prod\limits_{p\mid q}(1+p^{-1})^{-1}q^{\gamma}+O\left( \sum\limits_{p\mid q}(1-p^{-\frac{1}{2}}) ^{-1} q^{\gamma-\frac{1}{2}}\right)\\ &+O\left(q^{\frac{7}{13}\gamma+\frac{4}{13}}\prod\limits_{p\mid q}(1-p^{-\frac{1}{2}})^{-1}\log q\right)+O\left( q^{\frac{3}{4}}d^{3}(q)\log q \right)+O(q^{ \gamma -\frac{1}{6}} d^{2}(q)\log^3 q). \end{align}

    We need the error terms to be smaller than the main term, so

    \begin{cases} \frac{7}{13}\gamma+\frac{4}{13} < \gamma ,\\ \frac{3}{4} < \gamma ,\end{cases}

    which means the range of c is (1, \frac{4}{3}) . The reason why the range of c is changed is that R(c; q) requires q large enough.

    In this paper, we generalize the Lehmer problem by considering the count of square-free numbers in the intersection of the Lehmer set and Piatetski-Shapiro sequence when q is an odd integer and large enough. By methods of exponential sum and Kloosterman sum, we study its asymptotic properties and give a sharp asymptotic formula as q tends to infinity.

    Based on this result, we will consider some distribution problems similar to the Lehmer problem with more special sequences, which is significant for understanding the distribution properties of those problems.

    Xiaoqing Zhao: calculations, writing and editing; Yuan Yi: methodology and reviewing. All authors have read and agreed to the published version of the manuscript.

    In preparing this manuscript, we employed the language model ChatGPT-4 for the purpose of grammatical corrections. It did not influence the calculations and conclusion in this paper.

    This work is supported by Natural Science Foundation No.12271422 of China. The authors would like to express their gratitude to the referee for very helpful and detailed comments.

    The authors declare that there are no conflicts of interest regarding the publication of this paper.



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