An additive divisor problem in Z[i]

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Datum:	2003
Hauptverfasser:	Savasrtu, O.V., Varbanets, P.D.
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Sprache:	English
Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2003
Schriftenreihe:	Algebra and Discrete Mathematics
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spelling	nasplib_isofts_kiev_ua-123456789-1552782025-02-09T14:06:23Z An additive divisor problem in Z[i] Savasrtu, O.V. Varbanets, P.D. 2003 Article 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 Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
format	Article
author	Savasrtu, O.V. Varbanets, P.D.
spellingShingle	Savasrtu, O.V. Varbanets, P.D. An additive divisor problem in Z[i] Algebra and Discrete Mathematics
author_facet	Savasrtu, O.V. Varbanets, P.D.
author_sort	Savasrtu, O.V.
title	An additive divisor problem in Z[i]
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]
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2003
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.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT savasrtuov anadditivedivisorprobleminzi AT varbanetspd anadditivedivisorprobleminzi AT savasrtuov additivedivisorprobleminzi AT varbanetspd additivedivisorprobleminzi
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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