An additive divisor problem in Z[i]

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Published in:	Algebra and Discrete Mathematics
Date:	2003
Main Authors:	Savasrtu, O.V., Varbanets, P.D.
Format:	Article
Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2003
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Cite this:	An additive divisor problem in Z[i] / O.V. Savasrtu, P.D. Varbanets // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 103–110. — Бібліогр.: 6 назв. — англ.

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spelling	Savasrtu, O.V. Varbanets, P.D. 2019-06-16T15:28:44Z 2019-06-16T15:28:44Z 2003 An additive divisor problem in Z[i] / O.V. Savasrtu, P.D. Varbanets // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 103–110. — Бібліогр.: 6 назв. — англ. 1726-3255 2001 Mathematics Subject Classification: 11N37; 11R42. https://nasplib.isofts.kiev.ua/handle/123456789/155278 en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics An additive divisor problem in Z[i] Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	An additive divisor problem in Z[i]
spellingShingle	An additive divisor problem in Z[i] Savasrtu, O.V. Varbanets, P.D.
title_short	An additive divisor problem in Z[i]
title_full	An additive divisor problem in Z[i]
title_fullStr	An additive divisor problem in Z[i]
title_full_unstemmed	An additive divisor problem in Z[i]
title_sort	additive divisor problem in z[i]
author	Savasrtu, O.V. Varbanets, P.D.
author_facet	Savasrtu, O.V. Varbanets, P.D.
publishDate	2003
language	English
container_title	Algebra and Discrete Mathematics
publisher	Інститут прикладної математики і механіки НАН України
format	Article
issn	1726-3255
url	https://nasplib.isofts.kiev.ua/handle/123456789/155278
citation_txt	An additive divisor problem in Z[i] / O.V. Savasrtu, P.D. Varbanets // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 103–110. — Бібліогр.: 6 назв. — англ.
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first_indexed	2025-11-26T15:29:30Z
last_indexed	2025-11-26T15:29:30Z
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fulltext	Jo ur na l A lg eb ra D is cr et e M at h.Algebra and Discrete Mathematics RESEARCH ARTICLE Number 2. (2003). pp. 103–110 c© Journal “Algebra and Discrete Mathematics” An additive divisor problem in Z[i] O. V. Savasrtu and P. D. Varbanets Communicated by V. V. Kirichenko Abstract. Let τ(α) be the number of divisors of the Gaus- sian integer α. An asymptotic formula for the summatory function ∑ N(α)≤x τ(α)τ(α + β) is obtained under the condition N(β) ≤ x3/8. This is a generalization of the well-known additive divisor problem for the natural numbers. 1. Introduction In 1927 A.E. Ingham [1] obtained the asymptotic formula for the number of solutions I(x) the diophantic equation u1u2 − v1v2 = 1 under conditions: u1, u2, v1, v2 ∈ N, u1u2 ≤ x. Obviously I(x) = ∑ n≤x τ(n)τ(n + 1), where τ(n) = ∑ n=ab 1 denote the number of ways n may be written as a product of two natural numbers. Ingham proved that I(x) = 6 π2 x log2 x + O(x log x). T. Estermann [2] improved this result in form I(x) = xP2(log x) + E(x), 2001 Mathematics Subject Classification: 11N37; 11R42. Key words and phrases: additive divisor problem; asymptotic formula. Jo ur na l A lg eb ra D is cr et e M at h.104 An additive divisor problem in Z[i] where P2(u) is a polynom a0u 2 + a1u + a2 and E(x) is an error term. Estermann gave E(x) ≪ xθ+ε, θ = 11 12 . The exponent θ was sub- sequently improved to 5 6 by D.R. Heath-Brown [3] and then to 2 3 by J.-M. Deshouillers and H. Ivaniec [4]. In 1994 Y. Motohashi [5] em- ployed powerful methods from the spectral theory of automorphic forms and obtained very precise result: I(x) = ∑ n≤x τ(n)τ(n + h) = x 2 ∑ i=0 (log x)i 2 ∑ j=0 cij ∑ d\|h (log d)j d + O ( x 2 3 +ε ) holds uniformly for 1 ≤ h ≤ x20/27. The purpose of this paper is to build the asymptotic formula for sum ∑ α∈Z[i] 0<N(α)≤x τ(α)τ(α + β) where τ(α) = ∑ δ\|α 1 is a number of divisors of a Gaussian integer α. Notations. Denote by Z the ring of Gaussian integers. We write N(α) = a2+b2, Sp(α) = 2a for α = a+bi ∈ Z[α]; ϕ(α) = N(α) ∏ p\|α (1− N(p)−1), p is prime divisor α; e(x) = exp(2πix) for the real number x; the Vinogradov symbol f ≪ g means f = O(g); ε is an arbitrary small positive number that is not necessarily the same at each occurrence; the constants implied by the O (or ≪) – notation depend at most on ε. 2. Statement of Result Let β be Gaussian integer and x be real positive number. By I(x, β) we denote the number of solutions in Gaussian integers of the equation α1α2 − α3α4 = β, N(α1α1) ≤ x. Theorem. For N(β) ≪ x3/8 and any ε > 0 the following formula I(x, β) = xP2(log x) + O(x 7 8 +ε) holds. Here P2(u) = A0u 2 +A1u+A2, Ai = Ai(β), i = 0, 1, 2, moreover Ai(β) are computable and 1 ≪ Ai(β) ≪ τ(β), A0(β) > 0. Jo ur na l A lg eb ra D is cr et e M at h.O. V. Savasrtu, P. D. Varbanets 105 3. Auxiliary Results Let δ0, δ be the Gaussian rationals (δ0, δ ∈ Q(i)) not necessarily integers. Let for Re(s) > 1 ζ(s, δ, δ0) = ∑ ω∈Z[i] ω 6=−δ e ( 1 2 Sp(δ0ω) ) N(δ + ω)−s. Lemma 1 (see [5],lemmas 1 and 3). The function ζ(s, δ, δ0) is entire function if δ0 6∈ Z[i]. For δ0 ∈ Z[i], ζ(s, δ, δ0) is holomorphic except at s = 1, where it has a simple pole and ζ(s, δ, 0) = π s − 1 + a0(δ) + a1(δ)(s − 1) + . . . where a0(δ) = { πE + 4L′(1, χ4) if δ ∈ Z[i], πE + 4L′(1, χ4) + ∑ β∈B N(δ + β) + b0(δ) if 0 < N(δ) < 1; E is the Euler constant, L′(s, χ4) = d dsL(s, χ4), L(s, χ4) is L-Dirichlet function with non-principal character mod 4; b0(δ) = −4 + O ( N1/2(δ) ) , B denotes the set {0,±1,±i}. Moreover, the functional equation π−sΓ(s)ζ(s, δ, δ0) = π−(1−s)Γ(1 − s)ζ(1 − s,−δ0, δ)e ( − 1 2 Sp(δ0δ) ) (1) holds. Let α, β, γ ∈ Z[i]. We define the Kloosterman sum for the ring of Gaussian integer K(α, β; γ) = ∑ ξ,ξ′(mod γ) ξ·ξ′≡1(γ) e ( 1 2 Sp ( αξ + βξ′ γ )) . Lemma 2. Let α, β, γ be Gaussian integers, γ 6= 0. Then the estimate \|K(α, β; γ)\| ≪ (N(γ)N((α, β; γ)))1/2τ(γ) (2) holds, (where (α, β; γ) is the greate common divisor of α, β, γ). Moreover, K(α, β; γ) = ∑ δ\|(α,β,γ) N(δ)K ( 1, αβ δ2 ; γ δ ) . (3) Jo ur na l A lg eb ra D is cr et e M at h.106 An additive divisor problem in Z[i] This lemma follow from a multiplicative property of K(α, β; γ) on γ and the Bombieri estimate of an exponential sum on the algebraic curve over the finite field. The formula (3) is a generalized Kuznetsov’s identity for Kloosterman sums. Lemma 3. Let α0, γ ∈ Z[i], (α0, γ) = β, N(β) < N(γ). Then for N(γ) ≪ x2/3+ε we have ∑ α≡α0(γ) N(α)≤x τ(α) = c0(α0, γ) x N(γ) log x N(β) + + c1(α0, γ) x N(γ) + O ( x1/2+εN(γ)−1/4 ) , where c0(α0, γ) = π2N(β)ϕ ( γ β ) N−1(γ)τ(β), c1(α0, γ) = π2 ∑ δ\|β  2E − 1 + 2 L′(1, χ4) L(1, χ4) + ∑ p\|γ/δ ∗ log N(p) N(p) − 1   ∏ γ\|γ/δ ∗(1 − N−1(p)). Proof. Without loss of generality we will consider only a case (α0, γ) = 1. We have for c = 1 + ε: ∑ α≡α0(γ) N(α)≤x τ(α) − ∑ α=α0+βγ β∈B τ(α) = = 1 2πi c+iT ∫ c−iT  F (s) − ∑ β∈B τ(α0 + βγ) N(α0 + βγ)s   xs s ds + O ( xc TN(γ) ) , (4) where F (s) = N(γ)−2s ∑ α1,α2(mod γ) α1α2≡α0(γ) ζ ( s, α1 γ , 0 ) ζ ( s, α2 γ , 0 ) = ∑ α≡α0(γ) α∈Z[i] τ(α) N(α)s . From lemma 1 we have the functional equation F (s) = π2(2s−1) N2s(γ) Γ2(1 − s) Γ2(s) Ψ(1 − s), where Ψ(s) = ∑ ω 1 N(ω)s ∑ αβ=ω Φ(α, β; γ), Jo ur na l A lg eb ra D is cr et e M at h.O. V. Savasrtu, P. D. Varbanets 107 Φ(α, β; γ) ∑ α1,α2(mod γ) α1α2≡α0(γ) e ( 1 2 Sp ( αα1 + βα2 γ )) . Moreover, F (0) = 0 if N(γ) > 1 and α 6≡ 0(mod γ). By lemma 1 we obtain G(s) = F (s) − ∑ β∈B τ(α0 + βγ) N(α0 + βγ)s ≪ { N(γ)−1+ε if Re(s) = 1 + ε, N(γ)1/2+εT 3 if Re(s) = −1 4 . (5) Applying Phragmen-Lindelöf principle we infer G(−ε + it) ≪ N(γ)1/5+εT 12/5+ε for \|t\| ≤ T. To deal with integral in (4) we move the segment of integration to Re(s) = −ε. By the theorem of residues we obtain ∑ α≡α0(γ) N(α)≤x τ(α) = ress=0 ( G(s) xs s ) + ress=1 ( G(s) xs s ) + + 1 2πi −ε+iT ∫ −ε−iT G(s) xs s ds + +O (xε) + O ( N(γ)1/5+εT 12/5+ε ) + + O ( x1+ε TN(γ) ) . (6) Further, ress=0 ( G(s) xs s ) = π2x log x N(γ) ∏ γ\|α (1 − N(γ)−1)+ + π2x N(γ) ∏ p\|α (1 − N(p)−1)  −1 + 2  E + L′(1, χ4) L(1, χ4) + ∑ p\|δ log N(p) N(p) − 1     , (7) ress=0 ( G(s) xs s ) = ress=0  − ∑ β∈B τ(α0 + βγ) N(α0 + βγ)s xs s   ≪ N(γ)ε. Observe that by lemma 2 ∑ αβ=ω \|Φ(α, β; γ)\| = ∑ αβ=ω \|K(α, βα0; γ)\| ≪ N(γ)1/2N((ω, γ))1/2τ(γ)τ(ω). Jo ur na l A lg eb ra D is cr et e M at h.108 An additive divisor problem in Z[i] Now by termwise integration and applying the Stirling formula for the gamma function and the method of stationary phase we get 1 2πi −ε+iT ∫ −ε−iT G(s) xs s ds = = ∑ ω 0<N(ω)≤Y π2 N(ω) ∑ αβ=ω Φ(α, β; γ) y3/8 4 √ 2/π e ( − 1 8 − 1 2π y1/4 ) · · ( 1 + O ( y−1/8 )) + O ( x1+ε TN(γ) ) + O (xε)+ + O   ∑ N(ω)>Y y−εT 1+4εN(γ)1/2+εN((ω, γ))1/2τ(ω)N(ω)−1   , (8) where Y ≤ X = ( 4 π )4 T 4N2(γ) x , y = π4xN(ω) N2(γ) . The assertion of the lemma follow from (4),(6)–(8) if we put T = x1/2N(γ)−3/4, Y = x1/3. 4. Proof of the theorem We start the proof of our theorem by observing that τ(α) = 2#{γ\|α; N(γ) ≤ x1/2} − #{γ\|α; N(α)x−1/2 ≤ N(γ) ≤ x1/2} whenever α, N(α) ≤ x. Hence ∑ N(α)≤x τ(α)τ(α + β) = = ∑ N(γ)≤x1/2     2      ∑ α≡β(γ) N(α−β)≤x τ(α) − 1      −        ∑ α≡β(γ) N(α−β)≤N(γ)x1/2 τ(α) − 1            = = 2 ∑ N(γ)≤x1/2 ∑ α≡β(γ) N(α)≤x τ(α) − ∑ N(γ)≤x1/2 ∑ α≡β(γ) N(α)≤N(γ)x1/2 τ(α) + O(x7/8+ε) Indeed, we have N(α − β) = \|α − β\|2 ≥ ∣ ∣\|α\|2 − \|β\|2 ∣ ∣ = N(α) − N(β) for N(α) ≥ N(β), Jo ur na l A lg eb ra D is cr et e M at h.O. V. Savasrtu, P. D. Varbanets 109 and N(α − β) ≤ \|α\|2 + \|β\|2 = N(α) + N(β). Therefore we carry an error in the asymptotic formula ≪ N(β)x1/2 ≪ x7/8 if we replace the condition N(α−β) ≤ x on the condition N(α) ≤ x (we take into account that N(β) ≤ x3/8). Now, by lemma 3 we obtain ∑ N(α)≤x1/2 ∑ α≡β(γ) N(α)≤x = ∑ δ\|β ∑ N(γ)≤x1/2N(δ)−1 (γ,β/δ)=1 ∑ α≡β(mod γδ) N(α)≤x τ(α) = = ∑ δ\|β ∑ N(γ)≤x1/2 N(δ) (γ,α0/δ)=1 { π2x N2(γδ) N(δ)ϕ(γ)τ(δ) ( log x N(δ) − 1 ) + + 2π2x N(γ) ∑ t\|δ  E + L′(1, χ4) L(1, χ4) + ∑ p\|γδ/t log N(p) N(p) − 1   ∏ p\|γδ/t (1 − N(p)−1    + + O     ∑ δ\|β ∑ N(γ)≤x1/2 N(δ) x1/2N(γδ)−1/4     . Using the equality ϕ(α) = N(α) ∏ p\|α (1 − N(p)−1) = N(α) ∑ δ\|α µ(δ) N(δ) we infer ∑ N(α)≤x (α,β)=1 ϕ(δ) N(δ) = ∑ N(α)≤x (α,β)=1 ∑ δ\|α µ(δ) N(δ) = ∑ N(δ)≤x (αδ,β)=1 µ(δ) N(δ) ∑ N(α)≤ x N(δ) 1 = = ∏ p\|β (1 − N(p)−1)    πx ∑ N(α)≤x (δ,β)=1 µ(δ) N2(δ) + O(x1/3)    = c0(β)x + O(x1/3). (11) where c0(β) = c0 ϕ(β) N(β) ∏ p\|β (1 − N(ρ)−2), c0 = const. Therefore ∑ N(γ)≤x1/2 N(δ) (γ,β/δ)=1 c0(β/δ) ( log x + 1 + O ( x1/3 )) . (12) Jo ur na l A lg eb ra D is cr et e M at h.110 An additive divisor problem in Z[i] Hence, from (10),(12), we get ∑ N(γ)≤x1/2 ∑ α≡β(γ) N(α)≤x τ(α) = x(a0(β) log2 x+ + a1(β) log x + a2(β)) + O(x7/8+ε). (13) Similarly ∑ N(γ)≤x1/2 ∑ α≡β(γ) N(α)≤N(γ)x1/2 τ(α) = x(b1(β) log x + b2(β)) + O(x7/8+ε). (14) From (9),(13),(14) we obtain the assertion of theorem. References [1] A.E. Ingham, Some asymptotic formulae in the theory of numbers, 2. J. London Math. Soc. 2(1927), 202-208. [2] T. Estermann, Über die Darstellungen einer Zahl als Differenz von zwei Producten, J. Reine Angew. Math. 164(1931), 173-182. [3] D.R.Heath-Brown, The divisor function d3(n) in arithmetic progression, Acta Arithm. XLVII.1 (1986), 29-56. [4] J.-M. Deshoillers and H. Ivaniec, An additive divisor problem, Proc. London Math. Soc (2) 26(1982), 1-14. [5] Y.Motohashi, The binary additive divisor problem, Ann. Sci. École Normale Supériure (4) 27 (1994), 529-572. [6] P.D. Vatbanets and P.A. Zarzycki, Divisors of the Gaussian integers in an Arith- metic Progression, J. Number Theory (2) 33(1989), 152-269. Contact information O. V. Savasrtu ul.Dvoryanskaya 2, Dept. of computer alge- bra and discrete mathematics, Odessa na- tional university, Odessa 65026 Ukraine E-Mail: prik@imem.odessa.ua P. D. Varbanets ul.Solnechnaya 7/9 apt.18, Odessa 65009 Ukraine E-Mail: varb@te.net.ua Received by the editors: 22.02.2003.

An additive divisor problem in Z[i]

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