DIVISOR FUNCTION τ 3 (ω) IN ARITHMETIC PROGRESSION

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1 Annales Univ. Sci. Budapest., Sect. Comp DIVISOR FUNCTION τ 3 ω IN ARITHMETIC PROGRESSION A.S. Radova and S.P. Varbanets Odessa, Ukraine Dedicated to Professors Zoltán Daróczy and Imre Kátai on their 75th anniversary Communicated by Bui Minh Phong Received March 31, 2013; accepted May 10, 2013 Abstract. We constructed the asymptotic formula over the ring of the Gaussian integers for summatory function of the divisor function d 3ω in an arithmetic progression Nω l mod q which is non-trivial for q x 2 7 ε 1. Introduction Let Z[i] be the ring of Gaussian integers and k 2, k N. We define the divisor function τ k w, w Z[i] as the coefficient of Nw s in the Dirichlet series Z k s = τ k w, Rs > 1. Nw s w Here, as usually, Nw = w 2 is a norm of a Gaussian integer w, Zs is the Hecke Z-function. Key words and phrases: Kloosterman sum, divisor function, asymptotic formula Mathematics Subject Classification: 11L05, 11N37, 11N60.

2 262 A.S. Radova and S.P. Varbanets The main point of this paper is to consider the case k = 3 and to construct an asymptotic formula of the sum D k x; l, q = τ k α, 1 l < q, l, q = 1 Nw l mod q, Nw x in particular to investigate the ranges of q and x for which this formula is nontrivial. The similar problem was considered over Z in the works of Deshoulieres and Iwaniec[3] and Heath-Brown[4]. Over the ring of integer elements of the quadratic extension of Q in [6] it has been obtained an asymptotic formula for T k x, w 0, γ = τ k w, w 0, γ Q d, d Z, w w 0 mod γ, Nw x where γ is a fix integer from Q d. In rational case Deshoulieres and Iwaniec and also Heath-Brown used Deligne s bound for the k-fold Kloosterman sum to estimate D 3 x; l, q. In our paper we use the norm Kloosterman sums for which it is obtained a non-trivial estimate see, section 3. This let us to construct an asymptotic formula for D 3 x; l, q over Z[i] that can be applied for investigation of asymptotic behavior for the sum τnwτ 3 Nw. Nw x Notations. We denote G := Z[i] the ring of the Gaussian integers G = { a + bi a, b, Z, i 2 = 1 }. For the designation of the Gaussian integers we shall use the Greek letters α, β, γ, ξ, η; a Gaussian prime number denote through p if p Z. For α Z[i] we put Spα = α + α = 2Rα, Nα = α α, where α denotes a complex conjugate with α; Spα and Nα we name a trace and a norm respectively of α from Qi into Q. The writing a Z q respectively, a G γ denotes that a Z respectively, α G and a respectively, α runs a complete residue system modulo q modulo γ. Analogous, a Z q respectively, a G γ denotes a Zrespectively, α G and runs a reduced residue system modulo q respectively, modulo γ. The writing denotes that the summation runs over the region C which SC describe extra. For A N or α G put ν p A = a, or ν pα = a if p a A or p a α.

3 Divisor function τ 3ω in arithmetic progression 263 Moreover, exp z = e z, e q z = e 2πi z q for q N; the Vinogradov symbol as in fx gx means that fx = Ogx. The abbreviations e q and e are equal and use depend on the length of certain formula. 2πi... q For z Z respectively, z G, z, p = 1 let z 1 be the multiplicative inverse modulo p m. The signs and mean that the summation product conducts by all the non-associated integer respectively, prime Gaussian numbers. 2. Preliminary results We begin this section with a few background definitions and facts. Let q be a positive integer, q > 1, and let χ q be a Dirichlet character modulo q. Determine the function χ on the ring of Gaussian integers G as χα : G C, χα := χ q Nα for any α G. It is clear that χ is a character of the group G q. Lemma 1. Let l, q be the positive integers, q > 1, l 0 mod q, and let Jl, q denote the number of the solutions of x 2 + y 2 l mod q. Then we have where Jl, q = El, qq p a q p is odd a 1 p { 1 χ 4p νpl,pa +1 χ 4 p a+1 + p b=a ν pl,p a χ 4 p a b, 1 if q is odd or q 2 mod 4; 1 if q 0 mod 4, ν 2 l > ν 2 q 2; El, q = 2 if q 0 mod 4, l 2 ν2l 1 mod 4; 0 if q 0 mod 4, l 2 ν2l 3 mod 4, moreover, the sign in production denotes that if ν p l ν p q then an appropriate multiplier in this production have to be substituted on p a 1 χ 4 p a b. b=0

4 264 A.S. Radova and S.P. Varbanets Proof. The statement of Lemma follows by multiplication at q of the function Jl, q and the equation Jl, p a = 1 p a p a x,y Z p a z=1 e p azx 2 + y 2 l with the subsequent application the formula for the square of Gaussian sum Gh, p a = e p ahx. x Z p a Lemma 2. A non-principal character χ q produces a non-principal character χ if q 0 mod 4. For q 0 mod 4 it has only one non-principal character χ q producing the principal character χ 0 of group G q. Proof. Using the standard representation of a Dirichlet character, it s enough to prove only the case q = p n, p is a prime number. Every rational number a has a norm in Qi that equal to a 2, and hence, χa = χ q a 2, and therefore the equation χa = 1, a, q = 1 can be hold only if χ q a = ±1. Consequently, only the real characters χ q can produce the principal character χ 0. The basic modulo of the Dirichlet character is an equal to 2 m P, where m = 0, 1, 2, 3, and P = 1 or is a square-free odd number. So we should consider only the case of real characters with the basic modulus 2, 4, 8, p. If p > 2 be a prime number, for every b Z, b is a quadratic non-residue mod p, there exists α G such that Nα b mod p see, Lemma 1, and hence, for only one non-principal real character χ p x := is the Legendre symbol we have χα = χ p b = 1, i.e. χ is a non-principal character. For the modulo 2 we have only the principal character. For the modulo 4 a non-principal character χ 4 generates the principal character χ because Nα = 1 mod 4 if Nα, p = 1. At last, modulo 8 we have only one the non-principal character χ 8 which inducing χ 4. So, if q 0 mod 4, only the principal character χ q,0 produces the principal character χ 0, and for q 0 mod 4 we have only one character χ q inducing the principal character χ 0. That completes the proof of lemma. Lemma 3. Let p > 2 be a prime number, a 1, a 2, b 1, b 2, c, m Z, a 1, b 1, a 2, b 2, p = 1, m 1. Then for the exponential sum S := e p ma 1 x + a 2 y + b 1 x 2 + b 2 y 2 + pcxy x,y Z p m we have p m e p ma 1 a 1 + A 2 a 2 + B 1 b 1 + B 2 b 2 + C 1 c if m is even, S = p m+1 2 e p mc 1 a 1 + C 2 a 2 + D 1 b 1 + D 2 b 2 + C 2 c if m is odd x p x p

5 Divisor function τ 3ω in arithmetic progression 265 where A i, B i, C i, D i, C i Z, i = 1, 2. Proof. First, we suppose that m = 2n. Putting we infer S = x = x 0 + p n u, y = y 0 + p n v, x 0, y 0 Z p n, u, v Z p n e p 2na 1 x 0 + a 2 y 0 + b 1 x b 2 y0 2 + pcx 0 y 0 x 0,y 0 Z p n e p na 1 u + a 2 v + 2b 1 ux 0 + b 2 vy 0 + cpuy 0 + vx 0. u,v Z p n The summation over u, v shows that the inner sum vanishes if at least one of congruence 2.1 a 1 + 2b 1 x 0 + pcy 0 0 mod p n a 2 + 2b 2 y 0 + pcx 0 0 mod p n violates. Since a 1, a 2, b 1, b 2, p = 1 we have only one pair x 0, y 0 Z p m 2 such that 2.1 holds. Thus we have S = p 2n e p 2na 1 x 0 + a 2 y 0 + b 1 x b 2 y pcx 0 y 0, moreover, x 0, y 0 are linear combinations at a 1, a 2, b 1, b 2 with the coefficients from Z p 2n. Now let m = 2n + 1. Putting x = x 0 + p n u, y = y 0 + p n v, x 0, y 0 Z p n, u, v Z p n+1, modulo p 2n+1 we have and hence, S = x 0,y 0 Z p n e p n+1 u,v=0 x 2 x p n x 0 u + p 2n u 2, y 2 y p n y 0 v + p 2n v 2, xy x 0 y 0 + p n x 0 v + y 0 u + p 2n uv, a 1 x 0 +a 2 y 0 +b 1 x 2 0 +b 2 y2 0 +pcx 0 y 0 p 2n e a 1 u+a 2 v+2b 1 x 0 u+b 2 y 0 v+p n b1 u 2 +b 2 v 2 +pcx 0 v+y 0 u p n+1.

6 266 A.S. Radova and S.P. Varbanets The inner sum does not turn into zero only if pcx 0 + a 2 + 2b 2 y 0 0 mod p n, 2.2 pcy 0 + a 1 + 2b 1 x 0 0 mod p n. From 2.2 it follows that there is only one pair x 0, y 0 Z 2 p n 1 for which x 0 a 1 2b 1 + px 0, y 0 a 2 2b 2 + py 0 mod p n. In such case the inner sums over u and v are the Gaussian sums mod p n+1. Consequently, we obtain S = p m+1 2 e p md 1 x 0 + D 2 y 0 + E 1 x E 2 y 2 0, where D j, E j are the linear combinations at a 1, a 2, b 1, b 2 with the coefficients from Z p 2n+1. Now, we remind the some necessary information about the Hecke Z-function. Let δ 0, δ 1 Qx and let s C, Rs > 1. We define the Hecke Z-function by the absolute convergent series Z m s; δ 0, δ 1 = ω 4mi arg ω e Nω + δ 0 s e2πirδ1ω, where δ 0, δ 1 Qi, m Z. In the case m = 0, δ 0 = δ 1 = 0 we write Zs instead Zs; 0, 0. Lemma 4. For m 0 or m = 0 and δ 1 is not a Gaussian integer, Z m s; δ 0, δ 1 is an entire function. If m = 0 and δ 1 G, the function Z m is a holomorphic except as s = 1, where it has simple pole with a residue π. Moreover, the functional equation π s Γ2 m +sz m s; δ 0, δ 1 = π 1 s Γ2 m +1 sz m 1 s; δ 1, δ 0 e 2πiRδ0,δ1 holds in all the cases. For the proof in the case δ 0 = δ 1 = 0 see[5]. The proof in other cases is similar. Corollary 1. In the domain 1 4 Rs = σ 2, Is = t 3 the following estimates hold: log 4 m 2 + t 2 if 1 σ 2; m 2 + t 2 1 σ 3 log 4 m 2 + t 2 1 if 2 σ 1; Z m s; 0, 0 m 2 + t σ 18 log 4 m 2 + t 2 if 0 σ 1 2 ; m 2 + t 2 1 2σ if 1 4 σ 0.

7 Divisor function τ 3ω in arithmetic progression 267 This assertion follows from Lemma 4, an estimate Z m s, 0, 0 on half-line see, 6.16, [1] and the application of the Phragmen-Lindelöf theorem for an analytic functions in the strip 1 4 Rs Norm Kloosterman sums Let α 1,..., α n G, q > 1 be a positive integer. Denote, 3.1 Kn := K n α 0, α 1,..., α n ; q = SC e 2πiR α 0 ξ 0 + +αnξn q, where C := {ξ j G q, i = 0, 1,..., n, Nξ 0 ξ n 1 mod q}. K n we will call the n-fold norm Kloosterman sum. It s easy to check that for q 1 q 2 = q, q 1, q 2 = Kn α 0,..., α n ; q = K n α 0 q 2,..., α n q 2; q 1 K n α 0 q 1,..., α n q 1; q 2, where q 1 q 1 1 mod q 2, q 2 q 2 1 mod q 1. This property reduces our problem to compute K n a 0,..., a n ; q for prime power modulus q = p m. For a prime number p and the Gaussian integer α denote 3.3 r α = maxp r α = minν p Rα, ν p Iα. r In order to estimate K n α 0,..., α n ; p m it is sufficient to study the case minr α0,..., r αn = 0, i.e. at least one from α 0,..., α n does not divide on p. But then, putting α j = a j + ib j, j = 0, 1,..., n, we have GCDa 0,..., a n, b 0,..., b n, p = 1. We consider the case n = 2. First, we assume that m = 1. Let p 1 mod 4. Denote ξ j = x j + iy j, j = 0, 1, 2. Then from 3.1 it follows 3.4 K2 = SC e p a 0 x 0 + a 1 x 1 + a 2 x 2 b 0 y 0 b 1 y 1 b 2 y 2, where C := { x j, y j Z p : 2 x 2 j + y2 j }. 1 mod p j=0 Let ε 0 be a solution of congruence x 2 1 mod p.

8 268 A.S. Radova and S.P. Varbanets Put for every j = 0, 1, 2 u j = x j + ε 0 y j, v j = x j ε 0 y j. In view of ξ j G p it follows that Nξ j = x 2 j + y2 j 0 mod p and thus u j v j x 2 j + y2 j 0 mod p. Hence, u j, v j Z p. Furthermore, x j = 2 u j + v j, y j = 2 ε 0 u j v j, where mod p m. Consider three cases: i α j 0 mod p, j = 0, 1, 2; ii α 0 0 mod p, α 1 0, α 2 0 mod p; iii α 0 α 1 0 mod p, α 2 0 mod p. For the first case we have K 2 α 0, α 1, α 2 ; p = 2 e p A j u j + B j v j, SC j= C := u j, v j Z p, j = 0, 1, 2; u j v j 1 mod p. It is obvious that A j 2 a j + ε 0 b j mod p, B j 2 a j ε 0 b j mod p, and hence, A j B j 4 a 2 j + b2 j 4 Nα j mod p. But then A j, p = B j, p = 1. By 3.5 and the estimate of P. Deligne[2], we have 3.6 K 4p 5 2. j=0 For the second case we obtain K 2 α 0, α 1, α 2 ; p = e p u 0,v 0 Z p SC j=1 2 A j u j + B j v j, where C := u j, v j Z p, j = 1, 2; 2 u j v j u 0 v 0 mod p. j=1

9 Divisor function τ 3ω in arithmetic progression 269 For every collection of u 0, v 0 it is p 1 2 such collections at all we have the norm Kloosterman sum K 1 which has been estimated in [9]. Thus, in the case α 0 0 mod p, we obtain 3.7 K 2 0, α 1, α 2 ; p 2p 1 2 p 3 2 2p 7 2. At last, in the third case, we find at once 3.8 K 2 0, 0, α 2 ; p p 1 4 max a Z p x G p Nx a mod p Let now p 3 mod 4. In this case G p is a finite field F p 2, G p = p 2. Moreover, for any u, v Z u + iv p u p iv p u iv mod p, e p R x 2p 9 2. p and so 2Ru + iv T ru + iv mod p, where T r is a trace from F p 2 F p. So, the investigated sum K 2 coincide with the sum S 1 V, α = T rα X m, m, p = 1, p X V 1 e p into where α = α 0, α 1, α 2 F 3 p, X = x 2 1, x 2, x 3 F p, α X = 2 α i x i, V 1 is an algebraic manyfold produced by a polynomial x 0 x 1 x 2 1 over F p 2. Hence, applying to above the result of P. Deligne, we obtain S 2 V, α n + 1 lp l, where l is a number of α i under condition α i 0 mod p. Moreover, from the representation S 2 V, α by way of characteristic roots of the Riemann zeta-function of algebraic manifold over finite field we conclude S 2 1V, α 3S 2 V, α and thus we have 3.9 K 2 3p 5 2 +l. The case p = 2, m = 1 is trivial. Let m > 1. i=0

10 270 A.S. Radova and S.P. Varbanets There is no loss of generality in assuming that α 0 p. Thus it follows that a 0, b 0, p = 1. Let ξ 0 = x 0 + iy 0. By the definition of K 2, we may write 3.10 where K 2 = ξ j G p m j=0,1,2 = 1 p m SC e p m e p m 2 j=0 Rα j ξ j 1 p m 1 p m e p mknξ 0, ξ 1 ξ 2 1 = k=0 knξ 0 ξ 1 ξ a j Rξ j b j Iξ j, j=0 C := {k Z p m, ξ j G p m, j = 0, 1, 2}. Note, that the summation over k in 3.10 gives zero if Nξ 1, Nξ 2 are not coprimes to p. Therefore, we have K 2 = 1 p m ξ 1,ξ 2 G p m p m 1 k=0 x 0,y 0 Z p m kx 2 0 +y2 0 Nξ 1 Nξ a j Rξ j b j Iξ j e 2πi j=0 p m. Now, taking into account that the summation over x 0 or y 0 gives zero if k 0 mod p, we deduce K 2 = 1 p m e p m k e p m Rα 1 ξ 1 + α 2 ξ 2 k Z 3.11 p m ξ 1,ξ 2 G p m e p mf k, ξ, x 0,y 0 Z p where F k, ξ = knξ 1 ξ 2 x y0 2 + a 0 x 0 + b 0 y 0. The inner sum over x 0, y 0 in 3.11 is the product of two classical Gaussian sums, and, hence e p mknξ 1 ξ 2 x y0 2 + a 0 x 0 b 0 y 0 = 3.12 x 0,y 0 Z p m = e p m 4kNξ 1 ξ 2 a b p 1 2 p m. Here, as always A denotes the multiplicate inversive for A modulo p m if A, p = 1. We continue the calculation of K 2. From 3.11, 3.12 we infer 3.13 K2 = k Z p m 1 p 1 2 ep m k ξ 1,ξ 2 G p m e p mf 1 k, ξ,

11 Divisor function τ 3ω in arithmetic progression 271 where F 1 k, ξ = Put m 1 = [ m 2 2 a j Rξ j b j Iξ j 4 k a b 2 0Nξ 1 ξ 2. j=1 ], ξj = η j + p m1 ζ j, η j G p m, ζ j G p m m 1, 3.14 η j = x j + iy j, ζ j = u j + iv j, x j, y j Z p m 1, u j, v j Z p m m 1, j = 1, 2. Then 3.15 Nξ j = Nη j 1 2p m1 x j u j + y j v j Nζ j. Consequently, by , we obtain 3.16 where 3.17 K 2 = k Z p m e p m k 2 j=1 F 2 k, x, y = H 0 + p m1 H 1, x j,y j Z p m 1 x 2 j +y2 j,p=1 e p m 1 Rα j η j e p mf 2 k, x, y, u j,v j Z p m m 1 H 0 = 4 k a b 2 0x y1x y2 2 := 4 k a b 2 0D, 2 H 1 = 2 k a b 2 0Dx j u j + y j v j + a j u j b j v j. j=1 The summation over u j, v j in 3.16 gives zero if it disturbs at least one of the congruences 2ka j Da b 2 0x j 0 mod p m m1 3.18, j = 1, 2 2kb j + Da b 2 0y j 0 mod p m m1 If 3.18 holds, then the summation over u j, v j gives p 2m m1. Let min r α1, r α2 = 0 see, notation 3.3. In such case we have a 1, a 2, b 1, b 2, p = 1. Let a 1, p = 1. From the congruence ka 1 Da b 2 0x 1 0 mod p it follows that x 1, p = 1 and Da b 2 0 ka 1 x 1 mod p m m1. Thus from 3.18 we infer 3.19 x 2 a 1a 2 x 1, y 1 a 1b 1 x 1, y 2 a 1b 2 x 1 mod p m m1.

12 272 A.S. Radova and S.P. Varbanets By 3.16, 3.17, 3.19 and Lemma 1, we lead the estimate for K 2 to an exponential sum over k K 2 p 2m e p m k + Ak 5m 2p 2. k Z p m In the case min r α1, r α2 = r > 0 the system 3.18 has no solutions under assemption x 2 j + y2 j, p = 1, j = 1, 2 and, consequently, the sum K 2 is zero. The same estimates hold for K 2 if p = 1 + i. So, we proved our statement. Lemma 5. Let α 0, α 1, α 2 be the Gaussian integers, min r α0, r α1, r α2 = 0, and q > 1 be a positive integer, q = q 1 q 2, q 1, q 2 = 1, q 1 is square-free, q 2 is square-full. Then moreover, K 2 α 0, α 1, α 2 ; q = K 2 α 0 q 2, α 1 q 2, α 2 q 2 ; q 1 K 2 α 0 q 1, α 1 q 1, α 2 q 1 ; q 2, K 2 α 0 q 2, α 1 q 2, α 2 q 2 ; q 1 2 ωq1 p q 1 p 5 2 +lpα1,α2, where 0 if r α1 = r α2 = 0; l p α 1, α 2 = 1 if r α1 = 0, r α2 = 1; 2 if r α1 = r α2 = 1; 3 ωq2 q if min r α1, r α2 = 0 K 2 α 0 q 1, α 1 q 1, α 2 q 1 ; q 2 for all p q 2 ; 0 else. 4. Main results First we will suppose that l, q = 1 and l is a norm residue modq. For Rs > 1 we denote 4.1 F s; l; q := τ 3 α Nα s, α G SC

13 Divisor function τ 3ω in arithmetic progression 273 where C : {α G : Nα l mod q, Nα x}; 4.2 F s; l, q = F s; l, q Al l s, where It is clear, that 4.3 F s; l, q = α 1,α 2,α 3 G q Nα 1,α 2,α 3 l q Al = τ 3 α. α Nα=l Z s; α 1 q, 0 Z s; α 2 q, 0 Z s; α 3 q, 0 Nq 3s. Hence, be a simple generalization of the well-known Perron s formula for the Dirichlet series on an arithmetic progression see,[8] we have 4.4 T x; l, q := Nα l mod q Nα x + O x c T qc 1 3 τ 3 α = 1 2πi + Ox ε, c+it c it F s; l, q xs s ds+ where c > 1, 1 < T x are the parameters to be chosen later. Now we obtain by moving the path of integration to the line Rs = b, b > 0, 4.5 T x; l, q = Nα l mod q Nα x + 1 2πi 1 2πi b+it b it c it b it τ 3 α = res s=1 + res s=0 F s; l, q xs s ds + 1 2πi F s; l, q xs s ds + O F s; l, q xs s c+it b+it + F s; l, q xs s ds x c T qc O x ε. For the calculation of integrals in 4.5 we consider the function F s; l, q in the strip ε Rs c. It is obviously that 4.6 F c + it; l, q Nα l mod q Nα>q τ 3 α Nα c q 1+ε, ε > 0.

14 274 A.S. Radova and S.P. Varbanets On the line Rs = ε we apply the functional equation for Zs; δ, 0 and then obtain 4.7 6s π F s; l, q = π 3 Γ 3 1 s q Γ 3 s 3 e q R α j ω j Nω 1 ω 2 ω 3 1+s Al l s. α 1,α 2,α 3 G q ω 1,ω 2,ω 3 G Nα 1α 2α 3 l q By the absolute convergence of the series at ω 1, ω 2, ω 3 we may write j=1 4.8 F s; l, q = π 3 π q δ G δ q 6s Γ 3 1 s Γ 3 s ω G ω,q=δ Nω 1+s ω 1,ω 2,ω 3 G ω 1ω 2ω 3=ω K 2 ω 1, ω 2, u 0 ω 3 ; q, where Nu 0 l mod q. Putting δ = δ 1 δ 2 δ 3 and ω = δω = δ 1 ω 1δ 2 ω 2δ 3 ω 3 and applying Lemma 6 we have for s = ε + it: 4.9 F s; l, q q 6ε νq Γ 3 1 s Γ 3 s q 3 2 6ε 2 νq t ε. Hence, by and the Phragmen-Lindelöf theorem we infer for ε Rs c, Is F s; l, q q 3c 5σ 5ε 2c+ε t 3c σ c+ε. Thus trivially we have 4.11 c±it b±it F s; l, q xs s ds q 3 2 6b 2 νq T 2+6b + x T q 1 b. Moreover, by Stirling s formula and a simple transformation, we obtain 4.12 = δ q ω G δ G ω,q=δ b+it 1 2πi b it F s; l, q xs s ds = Nω 1 Φω Iω + O T 2+6b xnω q 6 b 6,

15 Divisor function τ 3ω in arithmetic progression 275 where Φω = ω 1ω 2ω 3=ω K 2 ω 1, ω 2, u 0 ω 3 ; q, Nu 0 l mod q 4.13 Iω = y b 2π 1 2πi T T 0 t 2+6b + O T 3+6b 0 y b, where y = π6 q 6 xnω, T 0 > 1 and will select later. In the last integral put t = y 1 6 t 1, ft = tlog t y it 6 e itlog t 1 + y it 6 e itlog t 1 gtdt+ y = y 1 6 t 1 log t 1 1. Since, f t 1 = 0 only if t 1 = 1, and f 1 = y 1 6 > 0, we [ conclude, that ] the integral Iω has only one stationary point t 1 = 1 if 1 y 1 6 T 0, y 1 6 T. Hence, for Nω > T 6 q 6 π 6 x, an integration by parts yields estimate Iω y b min T 5 Nωx 2 +6b, T 2+6b log T 6 q 6 log T. For Nω T 6 q 6 π 6 x, by the method of stationary phase see, [7]: Theorem 1.4, p.162 we deduce Iω = 1 1 π 2π π sin 4 y 1 6 y a1 y a2 y O y O T 2+6b + O T0 3+6b with the absolute constants a 1, a 2 in symbol O. So, to sum up, we have obtained from 4.4, 4.5, the following relation 4.16 T x; l, q = res s=1 res s=0 sin + O π 4 π 6 xnω q 6 xnω q F s; l, q xs + s δ G π 5 2 q 5 2 xnω 5 12 δ q ω,q=δ Nω X + ΦωNω 1 π 2π O q T 2 x ε log T + O q 5 2 +ε T0 3+6b log T,

16 276 A.S. Radova and S.P. Varbanets where X = T 6 q 6 π 6 x. After all this preliminary work, it is straight-forward to prove the main result of this paper. Theorem 1. Let l, q be the positive integers, 1 l < q, l, q = 1. Then for x we have 4.17 Nω l mod q Nω x τ 3 α = x q 2 Il, q p q P 2 log x+ Np + x q 2 Il, q 1 1 P 1 log x+ Np p q + 12x q 2 Iq, l + O x 5 +ε 7, where Iq, l is determined by Lemma 1 and P j u are the polynomials of j th - degree with the computable coefficients, moreover these coefficients and the constant in the error term do not depend on x, l, q. Proof. Let χ be an arbitrary character and let χ 0 be a principal character, both from the group of characters Ĝq modulo q. The Hecke Z-function with a character χ is defined by the series For χ = χ 0 Zs, χ = χω, Rs > 1. Nω s Zs, χ 0 = 1 χ 1 0p Np s = 1 N s p Zs. p G Since, Zs = ζsls, χ 4, we have Zs, χ 0 = πϕq 1 4q 2 s 1 + πϕq 4q 2 p q p q log Np Np ϕq π q 2 4 E + L 1, χ 4 + a 1s 1 + = = πϕq 1 4q 2 s 1 + b 0,qχ 0 + a 1s 1 +, where E is the Euler s constant, Ls, χ 4 is L-function of Dirichlet with the

17 Divisor function τ 3ω in arithmetic progression 277 non-principal character χ mod 4, ϕq = q Np, p q b 0,q χ 0 = πϕq 4q 2 E + L 1,χ 4 L1,χ + log Np 4 Np 1 p q. For an arbitrary character χ Ĝq we have 4.18 Zs, χ = εχ s 1 + b 0,qχ + b 1 χs 1 +, where Next, εχ = { πϕq 4q 2 if χ = χ 0 0 if χ χ 0. q 2s Z s, α 1 q, 0 = ϕq 1 χα 1 χ G q = 4 ϕq χ Ĝq χα 1 Zs, χ. α χα Nα s = Thus, by 4.18 q 2s Z s, α 1 q, 0 = π q 2 1 s b 0,q χχα 1 +. ϕq χ Ĝq = So, = res s=1 α 1,α 2,α 3 G q Nα 1α 2α 3 l q + 4π q 2 ϕ 2 q x { { res s=1 Nα 1α 2α 3 l q F s xs s = 6s xs q s 3 j=1 } Z s, αj q, 0 = π 3 q xp 6 2 log x + π2 x q 4 ϕq xp 1 log x b 0,q χ χ G q b 0,q χ 1 b 0,q χ 2 χ 1,χ 2 G q i=1,2, j=2,3, i j 3 χα j + j=1 χα i χα j.

18 278 A.S. Radova and S.P. Varbanets Taking into account that a summation over α 1, α 2, α 3 shows the terms with χ i χ 0 gives 0, we obtain 4.19 res s=1 F s xs = π3 ϕq 2 Iq, l s q 6 xp 2 log x + 3π2 Iq, l q 4 xp 1 log x+ + 12πIq, l q 2 x, ϕq where Iq, l is determined by Lemma 1, ϕq be totient Euler function in Z[i]. Moreover, 4.20 res F s xs = ζ0l0, χ = C 0 s=0 s is constant. Now, we see from 16, 18, 19 that the assertion of the Theorem is proved by choosing T 0 = T 1+6b 2+6b, b = 1 log x < ε, T = x 2 7 q 1. The asymptotic formula 4.17 is non-trivial if q x 2 7 +ε. Similar asymptotic formula for the divisor function τ 2 α under Nα l mod q have been obtained in [10] a non-trivial for q x ε. We claim that with a growth of k the divisor function τ k α in an arithmetic progression Nα l mod q has an asymptotic formula that is a non-trivial in still smaller range of a change q. The case l, q > 1 can be leaded easily to considered l, q = 1. Authors is very grateful to the referee for pointing out some inaccuracies that improved its quality. References [1] Coleman, M. D., The Rosser-Iwaniec seive in number fields, Acta Arith., , [2] Deligne, P., La conjecture de Weil, I, II, Publ. Math. IHES, I , ; II , [3] Deshouillers, I.-M. and M. Iwaniec, An additive divisor problem, Proc. London Math. Soc., , [4] Heath-Brown, D.R., The divisor function d k n in arithmetic progressions, Acta Arith., ,

19 Divisor function τ 3ω in arithmetic progression 279 [5] Hecke, E., Eine neue Art von Zeta Functionen und ihre Beziehungen zur Verteilung der Primzahlen, I, II, Math. Z., , ; , [6] Lay Dyk Thinh, On number of divisors in angles, Math. Sb., , in Russian. [7] Fedoryuk, M.V., Asymptotic, Integral and Series, M., Nauka, 1987 in Russian. [8] Varbanets, P. and F. Kovalchik, Distribution of norm of integer divisors in arithmetic progression, Ann. Univ. Such. Budapest, Sectio Math., V 1972, in Russian. [9] Varbanets, S., General Klosterman Sums over ring of Gaussian integers, Uk. Math., I , [10] Varbanets, P. and S. Varbanets, On divisor function over Z[i], Ann. Univ. Sci., Budapest, Sect. Comp., , Antonina Radova and Sergey Varbanets I.I. Mechnikov Odessa National University Odessa Ukraine radova as@mail.ru varb@sana.od.ua