The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables
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#A101 INTEGERS 25 (2025) THE HIGHER POWER MOMENTS OF THE COEFFICIENTS OF THE DEDEKIND ZETA FUNCTION OVER A POLYNOMIAL IN SIX VARIABLES Naveen K. Godara Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India [email protected] Prashant Tiwari Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab, India [email protected] Received: 4/10/25, Accepted: 10/26/25, Published: 11/5/25 Abstract Let Kbe a non-normal field of degree 3 over Q. Let ℓ≥2 be an integer. In this paper, we investigate the ℓth power moment of the coefficients attached to the Dedekind zeta function ζK(s) over a sequence. In particular, we consider the following: Sℓ(x) := X n=x2 1+x2 2+x2 3+x2 4+x2 5+x2 6≤x (x1,x2,x3,x4,x5,x6)∈Z6 aℓ K(n) and establish an asymptotic result, where ζK(s) = P∞ n=1 aK(n) nsis the Dedekind zeta function. 1. Introduction Let Kbe a number field of degree [K:Q] = dand OKbe its ring of integers. Then the Dedekind zeta function attached to Kis defined as ζK(s) := Y p⊆OK p=0 1−1 (Np)s−1 =X u⊆OK u=0 1 (Nu)s(1) for Re(s)>1, where the product is over non-zero prime ideals in OKand Nu denotes the absolute norm of u. For K=Q, the Dedekind zeta function ζK(s) is DOI: 10.5281/zenodo.17535302
INTEGERS: 25 (2025) 2 the Riemann zeta function. The function ζK(s) extends analytically to the entire complex plane except for a simple pole at s= 1, and the residue at s= 1 is given by the analytic class number formula lim s→1+(s−1)ζK(s) = 2r1(2π)r2hR ωp|DK|, where r1is the number of real embeddings of K, 2r2is the number of complex embeddings of K, h denotes the class number, Ris the regulator, ωis the number of roots of unity in K, and DKis the discriminant of K. The Dedekind zeta function satisfies a functional equation similar to the Riemann zeta function, ξK(s) = ξK(s−1), where ξK(s) := p|DK| 2r2πn/2s Γs 2r1 Γ(s)r2ζK(s), which is analytic in the whole complex plane except for the simple poles at s= 0 and s= 1. We rewrite Equation (1) as a Dirichlet series ζK(s) = ∞ X n=1 aK(n) ns,Re(s)>1,(2) where aK(n) denotes the number of integral ideals in Kwith norm n. The coefficients aK(n) of the Dedekind zeta function ζK(s) in Equation (2) satisfy the following properties: for all n≥1, we have aK(n)≥0; for all coprime integers mand n, aK(mn) = aK(m)aK(n); and for any ϵ > 0, aK(n)≤d(n)[K:Q]≪nϵ, where d(n) denotes the divisor function. Since the coefficients aK(n) are multiplicative, the Dedekind zeta function has the Euler product ζK(s) = ∞ X n=1 aK(n) ns=Y p1 + aK(p) ps+aK(p2) p2s+· · · +aK(pk) pks +· · · (3) for Re(s)>1. In 1949, Landau [13] investigated the first moment of aK(n) for a number field Kwith degree d≥2 and proved that X n≤x aK(n) = cKx+O(x1−2 d+1 +ϵ)
INTEGERS: 25 (2025) 3 for some constant cKdepending on K. Later, Chandrasekharan and Narasimhan [4] studied the second moment of aK(n) and proved that X n≤x a2 K(n)≪x(log x)d−1. For a Galois extension Kover Qof degree d > 1, Chandrasekharan and Good [3] proved that X n≤x aℓ K(n) = xPℓ(log x) + O(x1−2d−ℓ+ϵ), for every ϵ > 0 and integer ℓ≥2, where Pℓ(log x) denotes a suitable polynomial of degree dℓ−1−1. For a non-normal field extension Kover Qof degree 3 given by an irreducible polynomial x3+Ax2+Bx +Cof discriminant D < 0, Fomenko [6] investigated the first and second moments and proved X n≤x a2 K(n) = c1xlog x+c2x+O(x9/11+ϵ) and X n≤x a3 K(n) = xP3(log x) + O(x73/79+ϵ), where P3(log x) is a polynomial in log xof degree 4. L¨u [17] improved the error terms, and Liu [16] further improved the above-stated result of Fomenko. For a more recent development, see [7]. Let Kbe a non-normal field over Qof degree 3 given by an irreducible polynomial x3+Ax2+Bx +Cof discriminant D < 0. For a given natural number ℓ≥2, we consider the ℓth power moment of the Dedekind zeta function associated with K given by Sℓ(x) := X n=x2 1+x2 2+x2 3+x2 4+x2 5+x2 6≤x (x1,x2,x3,x4,x5,x6)∈Z6 aℓ K(n).(4) In this paper, we prove the following results. Theorem 1. Let Kbe a non-normal field over Qof degree 3given by an irreducible polynomial x3+Ax2+Bx +Cof discriminant D < 0. Then for any ϵ > 0, we have Sℓ(x) = (c1x3+c2x3log x+O(x25 9+ϵ)if ℓ= 2, x3P3(log x) + O(x79 27 +ϵ)if ℓ= 3, where c1, c2are some suitable constants and P3(log x)is a polynomial in log xof degree 4.
INTEGERS: 25 (2025) 4 Theorem 2. Let Kbe a non-normal field over Qof degree 3given by an irreducible polynomial x3+Ax2+Bx +Cof discriminant D < 0, and let ℓ≥4be an integer. Then for any ϵ > 0, we have Sℓ(x) = (x3Pℓ(log x) + O(x3−2 3ℓ+ϵ)for even ℓ≥4, x3Qℓ(log x) + O(x3−2 3ℓ+ϵ)for odd ℓ≥5, where Pℓ(log x), Qℓ(log x)are polynomials in log xof degree a0,ℓ +a3,ℓ −1. Here, a0,ℓ = 1 + ⌊ℓ 2⌋ X i=1 ℓ 2i(2i)! i!(i+ 1)! and a3,ℓ = ⌈ℓ 2−1⌉ X i=1 ℓ 2i+ 14(2i+ 1)! (i−1)!(i+ 3)!. 2. Preliminaries Throughout the paper, ϵdenotes an arbitrarily small positive constant, but not necessarily the same one as others, and any implied constant may depend on ϵ. Let Hk(SL(2,Z)) be the set of normalized Hecke eigenforms of even integral weight kfor the full modular group SL(2,Z). Let {λf(n)}n∈Nbe the normalized Fourier coefficients of the cusp form f∈Hk(SL(2,Z)) at infinity, i.e., f(z) = ∞ X n=1 λf(n)nk−1 2e2πιnz for all z∈H, where His the Poincar´e upper half-plane. The Fourier coefficients λf(n) are the Hecke eigenvalues of f, and these are real numbers. Also, λf(n) satisfies the following relation: λf(m)λf(n) = X d|gcd (m,n) λfmn d2.(5) As a consequence of Deligne’s [5] proof of Weil’s conjectures, we have |λf(n)| ≤ d(n)≪ϵnϵ for any ϵ > 0, where d(n) is the divisor function. The L-function associated with the normalized Hecke eigenform f(z) = ∞ X n=1 λf(n)nk−1 2e2πιnz is defined as L(f, s) = ∞ X n=1 λf(n) ns(6)
INTEGERS: 25 (2025) 5 for Re(s)>1. Since λf(mn) = λf(m)λf(n) for all positive integers mand nsuch that gcd(m, n) = 1, the L-function has the Euler product L(f, s) = Y p1−λf(p) ps+1 p2s−1 =Y p1−αp ps−11−βp ps−1 , where αp+βp=λf(p) and αpβp=|αp|=|βp|= 1. For a given Dirichlet character χof modulus N, the twisted L-function is defined as L(f⊗χ, s) := ∞ X n=1 λf(n)χ(n) ns for Re(s)>1.Note that both the L-functions L(f, s) and L(f⊗χ, s) have analytic continuations to the whole complex plane (see, [11, Section 7.2]). For j≥2,the jth symmetric power L-function of degree j+ 1 is defined as L(symjf, s) := Y p j Y i=0 1−αpj−iβpip−s−1= ∞ X n=1 λsymjf(n) nsfor Re(s)>1,(7) where λsymjf(n) is a real-valued multiplicative function. From Deligne’s bound [5], we have |λsymjf(n)| ≤ dj+1(n)≪ϵnϵ for any ϵ > 0, where dj(n) denotes the number of jfactors of a positive integer n. For j≥2, we define the twisted jth symmetric power L-function as L(symjf⊗χ, s) := ∞ X n=1 λsymjf(n)χ(n) nsfor Re(s)>1.(8) Also, both the L-functions L(symjf, s) and L(symjf⊗χ, s) have analytic continuations to the whole complex plane and satisfy nice functional equations (for more details, see [18, 19]). For a given Dirichlet character χof modulus N, the Dirichlet L-function is defined as L(s, χ) = ∞ X n=1 χ(n) nsfor Re(s)>1. Denote L(symjf, s) := (ζ(s) if j= 0, L(f, s) if j= 1 and L(symjf⊗χ, s) := (L(s, χ) if j= 0, L(f⊗χ, s) if j= 1.
INTEGERS: 25 (2025) 6 Let f= ∞ X n=1 λf(n)nk−1 2qn∈Hk(SL(2,Z)). The jth symmetric power L-function can be written as L(symjf, s) = Y p1 + λsymjf(p) ps+λsymjf(p2) p2s+· · · +λsymjf(pk) pks +· · · (9) for Re(s)>1. The coefficients λsymjf(n) of the Dirichlet series in Equation (7) and the Fourier coefficients λf(n) satisfy λf(pj) = λsymjf(p) = αj+1 p−βj+1 p αp−βp .(10) Let r6(n) denote the number of representations of a positive integer nby a polynomial x2 1+x2 2+x2 3+x2 4+x2 5+x2 6∈Q[x1, x2, x3, x4, x5, x6], i.e., r6(n) := #{x:= (x1, x2, x3, x4, x5, x6)∈Z6|n=x2 1+x2 2+x2 3+x2 4+x2 5+x2 6}. Note that r6(n) = 16 X m|n χn mm2−4X m|n χ(m)m2 := 16r(n)−4t(n),(11) where χis a non-principal Dirichlet character of modulus 4. Note that both r(n) and t(n) are multiplicative functions. Using Equation (11), the sum Sℓ(x) defined in Equation (4) can be expressed as Sℓ(x) = X n=x2 1+x2 2+x2 3+x2 4+x2 5+x2 6≤x aℓ K(n) = X n≤x aℓ K(n)r6(n) = 16 X n≤x aℓ K(n)r(n)−4X n≤x aℓ K(n)t(n).(12) Also, notice that r(p) = p2+χ(p), t(p) = 1 + p2χ(p).(13) From [6], we learn that ζK(s) = ζ(s)L(f, s), where fis a holomorphic cusp form of weight 1 for the congruence subgroup Γ0(|D|). This implies aK(n) = X m|n λf(m)
INTEGERS: 25 (2025) 7 and consequently, we get aK(p) = 1 + λf(p).(14) Recent years have witnessed an increasing focus on the average behaviour of arithmetical functions; refer to [2, 21] and the associated references for further details. 3. Auxiliary Results For a real number m, we write ⌊m⌋and ⌈m⌉for the floor and ceiling of m, respectively. Since aℓ K(p) = (1 + λf(p))ℓ,we have aℓ K(p) = ℓ X i=0 ℓ iλi f(p).(15) Following Equation (10) and [14], we write aℓ K(p) = a0,ℓ +a1,ℓλf(p) + ℓ X i=2 ai,ℓλsymif(p).(16) Let ℓ= 2mfor some m≥1. From [14, Lemma 7.1, Equation (38)], we have λℓ f(p) = λ2m f(p) = (2m)! m!(m+ 1)! + m−1 X r=1 (2m)!(2r+ 1)! (m−r)!(m+r+ 1)!!λsym2rf(p) + λsym2mf(p). (17) Let ℓ= 2m+ 1 be an odd integer for some m≥1. Then we have λ2m+1 f(p) = 2(2m+ 1)! m!(m+ 2)! + m−1 X r=1 (2m+ 1)!(2r+ 2) (m−r)!(m+r+ 2)!!λsym2r+1f(p) + λsym2m+1f(p). (18) From Equations (15), (16), (17), and (18), we get a0,ℓ = 1 + ℓ 2 X i=1 ℓ 2i(2i)! i!(i+ 1)! for even ℓ, 1 + ℓ−1 2 X i=1 ℓ 2i(2i)! i!(i+ 1)! for odd ℓ (19)
INTEGERS: 25 (2025) 8 and a3,ℓ = ℓ 2−1 X i=1 ℓ 2i+ 14(2i+ 1)! (i−1)!(i+ 3)! for even ℓ, ℓ−1 2 X i=1 ℓ 2i+ 14(2i+ 1)! (i−1)!(i+ 3)! for odd ℓ. (20) Next, we consider the Dirichlet series associated with aℓ K(n)r(n) and aℓ K(n)t(n) given by Rℓ(s) = ∞ X n=1 aℓ K(n)r(n) ns(21) and Tℓ(s) = ∞ X n=1 aℓ K(n)t(n) ns(22) for Re(s)>3. In this section, we find the L-decompositions of Rℓ(s) and Tℓ(s) involving known automorphic L-functions. Throughout, s∈Cdenotes the complex number s= γ+ιt, where Re(s) = γand Im(s) = t. Lemma 1. We have Rℓ(s) = (ζ(s−2)a0,ℓ L(sym3f, s −2)a3,ℓ Wℓ(s)Hℓ(s)for even ℓ≥4, ζ(s−2)a0,ℓ L(sym3f, s −2)a3,ℓ W′ ℓ(s)H′ ℓ(s)for odd ℓ≥5, where Wℓ(s) = Y 1≤t1≤ℓ t1=3 L(symt1f, s −2)at1,ℓ Y 0≤t2≤ℓ L(symt2f⊗χ, s)at2,ℓ and W′ ℓ(s) = Y 1≤t′ 1≤ℓ t′ 1=3 L(symt′ 1f, s −2)at′ 1,ℓ Y 0≤t′ 2≤ℓ L(symt′ 2f⊗χ, s)at′ 2,ℓ for some suitable constants at1,ℓ , at2,ℓ , at′ 1,ℓ, at′ 2,ℓ. Here, Hℓ(s),H′ ℓ(s)converge absolutely and uniformly in the right half-plane Re(s)>5 2with Hℓ(s),H′ ℓ(s)non-zero for Re(s)=3, where a0,ℓ and a3,ℓ are given in Equations (19) and (20). Proof. Since aℓ K(n)r(n) is a multiplicative function, Rℓ(s) has the Euler product Rℓ(s) = Y p1 + aℓ K(p)r(p) ps+ ∞ X k=2 aℓ K(pk)r(pk) pks (23)
INTEGERS: 25 (2025) 9 for Re(s)>3. Let ℓ≥4 be an even integer. Note that aℓ K(p)r(p)=(a0,ℓ +a1,ℓλf(p) + ℓ X i=2 ai,ℓλsymif(p))(p2+χ(p)) := c(p), where a0,ℓ and a3,ℓ are given in Equations (19) and (20). From the structure of c(p), we define the Dirichlet series associated with the coefficients c(n) as ∞ X n=1 c(n) ns =ζ(s−2)a0,ℓ L(sym3f, s −2)a3,ℓ Y 1≤t1≤ℓ t1=3 L(symt1f, s −2)at1,ℓ Y 0≤t2≤ℓ L(symt2f⊗χ, s)at2,ℓ , which is absolutely convergent in Re(s)>3, where at1,ℓ , at2,ℓ , at′ 1,ℓ, at′ 2,ℓ are some suitable constants. We also note that Y p1 + c(p) ps+· · · +c(pm) pms +· · · =ζ(s−2)a0,ℓ L(sym3f, s −2)a3,ℓ Y 1≤t1≤ℓ t1=3 L(symt1f, s −2)at1,ℓ ×Y 0≤t2≤ℓ L(symt2f⊗χ, s)at2,ℓ =: Gℓ(s) for Re(s)>3. Observe that c(n)≪ϵn2+ϵfor any ϵ > 0 and c(p) ps+c(p2) p2s+· · · +c(pm) pms +· · · ≪ ∞ X m=1 p(2+ϵ)m pmγ <1 for Re(s)≥3 + ϵ. Write P=aℓ K(p)r(p) ps+· · · +aℓ K(pm)r(pm) pms +· · · and Q=c(p) ps+· · · +c(pm) pms +· · · . From the above calculations, we observe that |Q|<1 in Re(s)≥3 +ϵ. Notice that,
INTEGERS: 25 (2025) 16 and Z5 2+ϵ+ιT 5 2+ϵ−ιT T2(s)xs sds≪x(5 2+ϵ)T5 4+ϵ. Therefore, 4X n≤x a2 K(n)t(n)≪x3+ϵ T+x5 2+ϵT5 4+ϵ. From Equation (12), we obtain S2(x) = x3P2(log x) + Ox3+ϵ T+Ox5 2+ϵT5 4+ϵ. To balance the error terms, we choose T=x2 9, which implies S2(x) = x3P2(log x) + O(x25 9+ϵ). For ℓ= 3: Consider the sum Pn≤xa3 K(n)r(n). From Lemma 3 and Perron’s formula, we have X n≤x a3 K(n)r(n) = 1 2πι Z3+ϵ+ιT 3+ϵ−ιT R3(s)xs sds +Ox3+ϵ T, where 1 ≤T≤x, for some Tto be chosen later. Using Cauchy’s residue theorem, we get X n≤x a3 K(n)r(n) = Ress=3R3(s)xs s +1 2πι(Z5 2+ϵ+ιT 5 2+ϵ−ιT +Z5 2+ϵ−ιT 3+ϵ−ιT +Z3+ϵ+ιT 5 2+ϵ+ιT )R3(s)xs sds +Ox3+ϵ T. Since R3(s) has a pole at s= 3 of order 5 coming out from ζ(s−2)4and L(sym3f, s− 2) (see Remark 1), we find 16 Ress=3R3(s)xs s=x3P3(log x), where P3(log x) is a polynomial of degree 4 in log x. Again, by using Lemmas 5-7,
INTEGERS: 25 (2025) 17 we get Z5 2+ϵ−ιT 3+ϵ−ιT +Z3+ϵ+ιT 5 2+ϵ+ιT )R3(s)xs sds≪Z1+ϵ 1 2+ϵ |R3(γ+ιT)|xγ+2 Tdγ ≪max 1 2+ϵ≤γ≤1+ϵ xγ+2T(26 21 +10 3+18 5+2)(1−γ)−1+ϵ ≪x3+ϵ T+x5 2+ϵT753 210 −1+ϵ and Z5 2+ϵ+ιT 5 2+ϵ−ιT R3(s)xs sds≪x(5 2+ϵ)T534 105 −13 14 +ϵ. Therefore, 16 X n≤x a3 K(n)r(n) = x3P3(log x) + Ox3+ϵ T+Ox5 2+ϵT534 105 −13 14 +ϵ. Now, consider Pn≤xa3 K3(n)t(n). Since T3(s) is analytic in the obtained region, we apply Perron’s formula and Cauchy’s residue theorem to obtain X n≤x a3 K(n)t(n) = 1 2πι(Z5 2+ϵ+ιT 5 2+ϵ−ιT +Z5 2+ϵ−ιT 3+ϵ−ιT +Z3+ϵ+ιT 5 2+ϵ+ιT )T3(s)xs sds +Ox3+ϵ T. On using Lemmas 5-7, we deduce Z5 2+ϵ−ιT 3+ϵ−ιT +Z3+ϵ+ιT 5 2+ϵ+ιT )T3(s)xs sds≪x3+ϵ T+x5 2+ϵT77 12 −1+ϵ and applying the Cauchy-Schwarz inequality, we get Z5 2+ϵ+ιT 5 2+ϵ−ιT T3(s)xs sds≪x(5 2+ϵ)T23 4+ϵ. Hence, 4X n≤x a3 K(n)t(n)≪x3+ϵ T+x5 2+ϵT23 4+ϵ. From Equation (12), we have S3(x) = x3P3(log x) + Ox3+ϵ T+Ox5 2+ϵT23 4+ϵ.
INTEGERS: 25 (2025) 18 We balance the error terms by choosing T=x2 27 , which further provides the desired asymptotic formula S3(x) = x3P3(log x) + O(x79 27 +ϵ). This completes the proof. Proof of Theorem 2. We only give the proof when ℓ≥4 is an even integer, and the other case follows similarly. Consider the sum Pn≤xaℓ K(n)r(n). From Lemma 1, we have Rℓ(s) = ζ(s−2)a0,ℓ L(sym3f, s −2)a3,ℓ Y 1≤t1≤ℓ t1=3 L(symt1f, s −2)at1,ℓ ×Y 0≤t2≤ℓ L(symt2f⊗χ, s)at2,ℓ Hℓ(s). From [14, Lemma 2.4], we learn that the degree of L(sym3f, s −2)a3,ℓ Y 1≤t1≤ℓ t1=3 L(symt1f, s −2)at1,ℓ is (3ℓ−a0,ℓ).From Lemma 1 and Perron’s formula, we have X n≤x aℓ K(n)r(n) = 1 2πι Z3+ϵ+ιT 3+ϵ−ιT Rℓ(s)xs sds +Ox3+ϵ T. Using Cauchy’s residue theorem, we get X n≤x aℓ K(n)r(n) = Ress=3Rℓ(s)xs s +1 2πι(Z5 2+ϵ+ιT 5 2+ϵ−ιT +Z5 2+ϵ−ιT 3+ϵ−ιT +Z3+ϵ+ιT 5 2+ϵ+ιT )Rℓ(s)xs sds +Ox3+ϵ T. From Remark 1, the order of the pole at s= 3 in Rℓ(s) is (a0,ℓ +a3,ℓ); therefore, we have 16Ress=3Rℓ(s)xs s=x3Pℓ(log x), where Pℓ(log x) is a polynomial in log xof degree (a0,ℓ +a3,ℓ)−1. We use results from Subsection 3.1 to estimate the integrals appearing in the proof of this theorem,
INTEGERS: 25 (2025) 19 from which one can easily find Z5 2+ϵ−ιT 3+ϵ−ιT +Z3+ϵ+ιT 5 2+ϵ+ιT )Rℓ(s)xs sds≪Z1+ϵ 1 2+ϵ |Rℓ(γ+ιT)|xγ+2 Tdγ ≪max 1 2+ϵ≤γ≤1+ϵ xγ+2T13 42 (a0,ℓ)+ 3ℓ 2−a0,ℓ 2(1−γ)−1+ϵ ≪x3+ϵ T+x5 2+ϵT13a0,ℓ+21(3ℓ−a0,ℓ) 84 −1+ϵ and Z5 2+ϵ+ιT 5 2+ϵ−ιT Rℓ(s)xs sds≪x(5 2+ϵ)T −13 14 +13a0,ℓ+21(3ℓ−a0,ℓ) 84 +ϵ . Therefore, 16 X n≤x aℓ K(n)r(n) = x3Pℓ(log x)+Ox3+ϵ T+Ox5 2+ϵT −13 14 +13a0,ℓ+21(3ℓ−a0,ℓ) 84 +ϵ. Now, consider the sum Pn≤xaℓ K(n)t(n). Since Tℓ(s) is analytic (as in Lemma 2) in the obtained region, we apply Perron’s formula and Cauchy’s residue theorem to get X n≤x aℓ K(n)t(n) = 1 2πι(Z5 2+ϵ+ιT 5 2+ϵ−ιT +Z5 2+ϵ−ιT 3+ϵ−ιT +Z3+ϵ+ιT 5 2+ϵ+ιT )Tℓ(s)xs sds +Ox3+ϵ T. Note that the degree of Y 1≤t2≤ℓ L(symt2f⊗χ, s −2)at2,ℓ is (3ℓ−a0,ℓ).Therefore, we obtain Z5 2+ϵ−ιT 3+ϵ−ιT +Z3+ϵ+ιT 5 2+ϵ+ιT )Tℓ(s)xs sds≪x3+ϵ T+x5 2+ϵT4a0,ℓ+6(3ℓ−a0,ℓ) 24 −1+ϵ and applying the Cauchy-Schwarz inequality, we get Z5 2+ϵ+ιT 5 2+ϵ−ιT Tℓ(s)xs sds≪x(5 2+ϵ)T3ℓ 4−1+ϵ.
INTEGERS: 25 (2025) 20 Hence, 4X n≤x aℓ K(n)t(n)≪x3+ϵ T+x5 2+ϵT3ℓ 4−1+ϵ. From Equation (12), we have Sℓ(x) = x3Pℓ(log x) + Ox3+ϵ T+Ox5 2+ϵT3ℓ 4−1+ϵ. We choose T=x2 3ℓto get Sℓ(x) = x3Pℓ(log x) + Ox3−2 3ℓ+ϵ, where Pℓ(log x) is a polynomial in log xof degree a0,ℓ +a3,ℓ −1. This completes the proof. Acknowledgement. The authors thank their affiliated institutions for support and the anonymous referees and managing editor Bruce Landman for their valuable comments and suggestions, which improved the clarity of the manuscript. The second author also acknowledges support from the NBHM postdoctoral fellowship. References [1] J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30(1) (2017), 205–224. [2] K. Chakraborty, N. K. Godara, and L. Vaishya, Shifted convolution sums of divisor functions associated with the symmetric power lifts of GL(2)-forms, Ramanujan J. 68 (2025), #A55. [3] K. Chandrasekharan and A. Good, On the number of integral ideals in Galois extensions, Monatsh. Math. 95(2) (1983), 99–109. [4] K. Chandrasekharan and R. Narasimhan, The approximate functional equation for a class of zeta-functions, Math. Ann. 152 (1963), 30–64. [5] P. Deligne, La conjecture de Weil. I, Inst. Hautes ´ Etudes Sci. Publ. Math. 43 (1974), 273–307. [6] O. M. Fomenko, Mean values associated with the Dedekind zeta function, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 350 (2007), 187–198. [7] N. K. Godara and P. Tiwari, On the general divisor problem of power moments of the coefficients attached to the Dedekind zeta function over a sparse sequence, Results Math. 80(5) (2025), #A125. [8] A. Good, The square mean of Dirichlet series associated with cusp forms, Mathematika 29(2) (1982), 278–295.
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