Annales Univ. Sci. Budapest., Sect. Comp. 46 (2017) 235–246
ON EXPONENTIAL SUMS INVOLVING THE DIVISOR FUNCTION OVER Z[i]
Antonina Radova and Sergey Varbanets (Odessa, Ukraine)
Dedicated to the memory of Professor Antal Iv´anyi
Communicated by Imre K´atai
(Received February 5, 2017; accepted March 30, 2017)
Abstract. We apply the van der Corput transform to investigate the sums of view P r(n)g(n)e(f(n))), where r(n) is the number of representations of n as the sum of two squares of integer numbers. Such sums have been studied by M. Jutila, O. Gunyavy, M. Huxley and etc. Depending on differ- ential properties of the functions g(n) and f(n) there have been obtained different kinds of error terms in bounds of the considered sums. In the special case, O. Gunyavy improved the result of M. Jutila in the problem on estimate the exponential sum involving the divisor function τ(n). We P a obtain the asymptotic formula of the sum τk(α)e q N(α) , k = 2, 3 over the ring of Gaussian integers which is an analogue of the asymptotic formulas obtained by M. Jutila and O. Gunyavy.
1. Introduction
In 1985 M. Jutila constructed an asymptotic formula for the sum
X an τ(n)e , q n≤N
Key words and phrases: Exponential sum, divisor function. 2010 Mathematics Subject Classification: 11A25, 11N56, 11N37. 236 A. Radova and S. Varbanets which is a special case of more general sum
X f(n) a(n)g(n)e , q n≤N where a(n) is a multiplicative function, and f(x), g(x) are real-value differen- tiable functions on [1, ∞). Such sums were studied in the works of M. Jutila[4] [5], A. Kr¨atzel[6],M. Haxley[3] etc. The asymptotic formula of M. Jutila for a(n) = τ(n) is nontrivial for q 2 −ε x 5 . O. Gunyavy[2] has extended the field of nontriviality for this formula 1 +ε up to q x 2 . The primary purpose of our paper is to obtain the analogical asymptotic formulas for divisor function τk(α) over the ring of Gaussian integers. Notations. We will frequently use the Landau and Vinogradov asymptotic notations. The big ”O” notation f(x) = O(g(x)) (equivalently, f(x) g(x)) means that there exists some constant C such that |f(x)| ≤ C|g(x)| on the domain in question. By f(x) g(x), we shall mean that g(x) f(x) g(x). We denote e2πix as e(x).
2. Main lemma
Consider real-valued functions f(x) and g(x), x ∈ [N,N1], N ≤ N1 ≤ 2N. We will suppose that the following conditions
(i) g0(x), f 00(x) are monotonic;
0 g(N) 1 0 0 0 (ii) g(x) g(N), g (x) N , N |f (N)| |f (x)| |f (N)|;
0 0 0 00 0 (3) |f (N)| (iii) |f (N)| |f (x) + 2xf (x)| |f (N)|, |f (x)| N 2 are true. We prove the following statement that allows to study the distribution of values of the divisor function over the ring of Gaussian integers.
Lemma. Let the functions f(x) and g(x) satisfy the conditions above, and let r(n) be the number of the representation of n by form n = u2 + v2, u, v ∈ Z. On exponential sums 237
Then we have X r(n)g(n)e2πif(n) = N≤n≤N ≤2N 1 √ X p 0 2πi(f(4ϕ(n))−2 nϕ(n)) = ωf r(n)g(4ϕ(n)) |ϕ (n)|e + 1 2 X≤n≤2X ε g(N) ε + O N + O(N |g(N)|)+ f 0(N) !! 1 +ε p 1 + O |g(N)| N 2 min |f 0(N)|, , p|f 0(N)| where −1 if f 0(N)(f 0(n) + 2Nf 00(N)) > 0, 0 0 00 ωf = i if f (N) > 0, f (N) + 2Nf (N) < 0, −i if f 0(N) < 0, f 0(N) + 2Nf 00(N) > 0, X = 2Nf 02(N), ε > 0 is an arbitrary small, and constatnts in symbols ”O” depend only on ε. Proof. Applying the asymptotic representation of summatory function r(n) (see, [6], Ch. VIII, formula (685)) ∞ X √ X r(n) √ r(n) = πx + x √ I (2π nx), n 1 n≤x n=1 where Iν (u), ν = 0, 1, 2,... is the Bessel function of kind one and ν-th order, we can write for arbitrary X > 1 X r(n)g(n)e(f(n)) =
N≤N2≤N1<2N
N1 Z d √ X r(n) √ = g(x)e(f(x)) πx + x √ I (2π nx) + dx n 1 N n≤X (1) ! N1 √ X r(n) √ + x √ I ( nx) g(x)e(f(x)) + n 1 n>X N
N1 Z d √ X r(n) √ + (g(x)e(f(x))) x √ I (2π nX)dx. dx n 1 N n>X 1 0 02 First we consider the case N f (N) 1. Put X = 2Nf (N). Estimat- ing the trigonometrical integral with respect to first derivation, we have Z g(N) (2) g(x)e(f(x))πdx . f 0(N) 238 A. Radova and S. Varbanets
From the estimate
r √ X r(n) √ x R(x) := x √ I (2π nx) Xε 1 + n 1 X n>X we obtain Xεg(N) R(x)g(x)e(f(x)) . f 0(N)
Next, since g0(x) is monotone function, then
N1 N1 Z Z r x g0(x)e(f(x))R(x)dx N ε |g0(x)| dx X (3) N N N1 N ε Z N εg(N) |g0(x)|dx . f 0(N) f 0(N) N
1 √ √ √ d 2 Taking into account that dx x I1(2π nx) = π nI0(2π nx) we have
N N1 X1 X Z √ τ(n)g(n)e(f(n)) = π r(n) I0(2π nx)g(x)e(f(x))dx+ n=N n≤X N
N1 (4) X r(n) Z √ √ + 2πi √ g(x)f 0(x)e(f(x)) xI (2π nx)dx+ n 1 n>X N N εg(N) N εg(N) + O = S + S + O , f 0(N) 1 2 f 0(N) say. 02 02 Select X = max {Nf (N),N1f (N1)}. By the formulas for the Bessel functions
r 2 πν π 1 I (x) = cos x − − 1 + O , x 1, ν = 0, 1,..., ν πx 2 4 x we at once derive that the critical points of subintegral functions in S1 and S2 1 lie beyond interval of integration if n 6∈ 2 X,X . Thus for such values of n an On exponential sums 239 integration by parts gives
N1 X Z √ r(n) I0(2π nx)g(x)e(f(x))dx 1 n≤ 2 X N
g(N) p 0 1 + g(N) Nf (N) log N, N 2 f 0(N) (5) N Z 1 X r(n) 1 √ 0 √ x 2 I (2π nx)g(x)f (x)e(f(x))dx n 1 n>X N N εg(N) + N εg(N)pNf 0(N). f 0(N)
For the other values of n we have
N1 X Z √ r(n) g(x)e(f(x))I0(2π nx)dx = 1 2 X N1 (6) πi 0 e 4 X r(n) Z g(x)f (x) √ = − √ 1 1 e(f(x) − nx)dx+ 4 4 2 1 n x 2 X The integral on the right-hand side in (6) will be calculated by the van der Corput method (see, [8], Ch. 4. Lemma 3 or [9]). We have 1 N N1 Z 1 Z4 1 g(4x)e(fn(x)) − 4 I(n) := x g(x)e(f(x))dx = 4 √ 1 dx = 2x 4 N 1 4 N (7) g(4ϕ(n))e(fn(ϕ(n))) g(N) = ωf + O + 1 p 00 1 00 ϕ(n) 4 |fn (ϕ(n))| N 4 (ϕ(N) − N)fn (ϕ(n)) g(N) + O , 1 00 N 4 Nfn (N) √ 0 where fn(x) = f(4x) − 2 nx, ϕ(n) is the solution of the equality fn(x) = 0, i.e. 16xf 02(x) = n. 240 A. Radova and S. Varbanets 1 Simple calculations show that for n ∈ 2 X,X √ r 00 00 n 1 00 n fn (x) = f (x) + 3 = 4xf (x) + 2 4x x (8) 4x 1 f 0(N) (f 0(N) + Nf 00(N)) > 0. N N So, from (1)-(8) we find N1 P r(n)g(n)e(f(n)) = n=N P p 0 p = ωf r(n)g(ϕ(n)) |ϕ (n)|e(f(ϕ(n)) − nϕ(n))+ 1 2 X X p 0 ωf = r(n)g(ϕ(n)) |ϕ (n)|e(−fe(ϕ(n))), where bar denotes the complex conjugate value, and fe(n) = −f(ϕ(n)) + ϕ(n)f 0(ϕ(n)). Its clear that the following relations 1 fe0(x) = 1, f 0(ϕ(x)) fe0(x) + 2xfe00(x) = 4[f 0(ϕ(x)) + 2ϕ(x)f 00(ϕ(x))]−1, 4xfe02(x) = ϕ(x). are true. 1 Hence, for 2 X ≤ n ≤ X g(ϕ(n))pϕ0(n) g(N)(f 0(N))−1. So, we have X X p 0 r(n)g(n)e(f(n)) = ωf r(n)g(ϕ(n)) |ϕ (n)|e(fe(ϕ(n)))+ 1 N≤n≤N1 2 X≤n≤X √ ! N + O (N εg(N)) + O N εg(N) . pf 0(N) On exponential sums 241 For f 0(N) < 0 we instead of sum P r(n)g(n)e(f(n)) consider the complex conjugate sum P r(n)g(n)e(−f(n)). So, lemma is proved. This lemma allows to prove the following statements. 3. Main results By an analogy with the work of M. Jutila[5] it can be shown that for (a, q) = = 1. We have a X 2πi an x a (9) A(x, ) = r(n)e q = π + R x, , q q q n≤x where for 1 ≤ N << x √ nx π cos 2π − 1 +ε ! a 1 √ X an q 4 x 2 4 2πi q (10) R x, = x q r(n)e 3 + O q 1 . q 4 2 n≤N n N The following theorem is an analogue of the Jutila theorem for divisor func- tion τ(n). For that we will use Main lemma proved above. Theorem 1. Let a, q be the positive integers, 1 ≤ a ≤ q, (a, 2q) = 1. Then 1 for an arbitrary 0 < ε < 2 we have a X a πx 1 +ε A(x; ) := r(n)e n = + O x 2 . q q q n≤x 1 −ε (This asymptotic formula is nontrivial for q ≤ x 2 ). Proof. We have K a X 2πi an X X 2πi an A(x; ) = r(n)e q = r(n)e q + q (11) n≤x k=0 2k≤n<2k+1 X 2πi an + r(n)e q , 2K a The function f(x) = q x suffices the conditions of Main lemma. In sums over n on the right side of the expression above we apply the Main lemma, and then sum over all k. We deduce r a X q 2πi− − 4q n ε q ε xa A(x; ) =ω r(n) e ( a ) + O x + O x . (12) q f a a q 2 n≤ a x q2 q2 (Here we take into account that in the Main lemma for our case ϕ(n) = a2 n). 0 0 00 Note that in this case ωf = 1 (because f (N) · (f (N) + 2Nf (N)) > 0). So, we obtain a q a2 4q q rxa A(x; ) = A x, − + O xε + O xε . q a q2 a a q Define a0, 1 ≤ a0 < a, from the congruence a0 ≡ −4q (mod a). Therefore, we can write a q a2 a q rxa A(x; ) = A x, 0 + O xε + O xε . q a q2 a a q By proceeding this reasoning we infer the sequence of equalities 2 2 q q xa a0 q xa0 −a ε q εp xa0 A 2 , = A 2 , + O x + O x a q a a0 q a0 a0 a ...... 2 2 q xai−1 ai q xai −ai−1 εq q εq xai A 2 , = A 2 , + O x + O x . ai−1 q ai−1 ai q ai ai ai−1 By virtue of the fact that ai ∈ N, a0 > a1 > . . ., and ai+1 ≡ −4ai (mod ai−1), we draw a conclusion that for the some index i0 will have ai0 = 1. It is clear that i0 ≤ n0, where n0 is an integer of Fibonacci number nearest to q. Hence, i + 0 log q xε. Now we obtain after log q steps the following relation log q log q 1 2 q q x X q X xaj+1 A x, = A , 1 + O xε + O xε = q 1 q a a j=0 j j=0 j πx ε 1 +ε = + O (qx ) + A x 2 . q The result of this theorem has larger region of nontriviality of the asymp- totic formula than region of nontriviality which can be obtained from (9)–(10). On exponential sums 243 With help of Theorem 1 we can study the distribution of the values of the divisor function τ(α) over the ring of Gaussian integers, where X τ(α) = 1. α1,α2∈Z[i] α1α2=α Theorem 2. Let α0, β be the Gaussian integers, (α0, β) = 1, and τ(α) be the 1 −ε divisor function over the ring of Gaussian numbers. Then for N(β) x 4 the following asymptotic formula X 2πiN( α0α ) x log x x 3 +ε 1 +ε τ(α)e β = C (β) +C (β) +O x 4 +O x 2 N(β) 1 N(β) 2 N(β) N(α)≤x −ε ε where Ci(β) are computable constants, N(β) Ci(β) N(β) , i = 1, 2, holds. Proof. Denote N(α0) = a, N(β) = q. Then, we have 2πiN αα0 2πi aN(α1)N(α2) 2πi amn P τ(α)e ( β ) = P e q = P r(m)r(n)e q = N(α)≤x α1,α2∈Z[i] mn≤x N(α1α2)≤x 2πi amn 2πi amn = 2 P P r(m) P r(n)e q − P P P r(m)r(n)e q = n≤ x 1 1 d|q (m,q)=d m d|q 2 2 1 m≤x n≤x m≤x 2 (m,q)=1 P P = 2 1 − 2, say. For (m, q) = d let us suppose m = m1d, q = q1d,(m1, q1) = 1. Then, from the Theorem 1, we get am1n P P P P 2πi q 1 = r(m1d) r(n)e 1 = d|q 1 n≤ x x 2 m1d m1≤ d " 1 !# 2 +ε ε P πx P r(m1d) P x x = + O + q1 = dq1 m1 m1d m1d d|q 1 1 x 2 x 2 m1≤ d m1≤ d (m ,q )=1 (13) (" 1 1 # s−1 P πx P Q χ4(d1) x = q res (ζ(s)L(s, χ4)) χ4(d1) 1 − ps s−1 + s=1 d d|q d1|d p| d1 1 ! ) 1 3 +ε 2 3 1 x 4 +ε −1−ε 2 +ε +O d + O x d + O qx = x log x 1 3 x 2 +ε 4 +ε = A1(q) q + A2(q) q + O x q + O x , 244 A. Radova and S. Varbanets −ε ε where A1(q), A2(q) be the computable constants, q Ai(q) q , i = 1, 2. As before, we obtain 1 P P P πx 2 1 +ε ε = r(m) + O x 4 + O (q1x ) = 2 q1 1 d|q 2 x m1≤ d (14) (m1,q1)=1 3 x 4 +ε ε = B(q) q + O x + O (q1x ) , where q−ε B(q) qε. Collecting out estimates(13)–(14) together, we obtain the desired result of theorem. Let τ3(α) be the number of representations of Gaussian integer α in form α = α1α2α3, αj ∈ Z[j]. We proved the following statement. Theorem 3. Let a and q be the positive integers, (a, q) = 1. Then for x → ∞ X aN(α) x τ (α)e = P (log x) + O xθ0 , 3 q q 2 N(α)≤x 2−θ where P2(u) is a polynomial of two degree with the fixed coefficients θ0 = 3−2θ , 1792 θ = 3615 . Proof. For 1 ≤ X < x we have X aN(α) X aN(α1α2α3) T (x; a, q) = τ (α)e = e = 3 3 q q N(α)=x α1,α2,α3 α1α2α3=α N(α1α2α3)≤x X X aN(α1α2)N(α3) = 1 e = α ,α q 1 2 α3∈Z[i] α1,α2=α x N(α3)≤ N(α1α2)≤x N(α1α2) X X aN(α) = τ(α) r(m)e m + x q N(α)≤X m≤ N(α) X X aN(α) + τ(α) r(m)e m = x q X Now the application of theorems 1 and 2 gives 1 +ε P aN(α) P π x x 2 τ3(α)e q = τ(α) q · N(α) + O N(α) + N(α)≤x N(α)≤X n + P r(m) C1(q) x log x − X log X + C2(q) x − X + x q m m q m m≤ X 3 1 o x 4 +ε x 2 +ε +O m + O m q . In the right side of the first sum above, we will use the following asymptotic formula (see, [1]) X π2 π π2γ π2 τ(α) = x log x + L0(1, χ ) + + x + O xθ , 16 2 4 8 16 N(α)≤x 1792 θ = < 0, 496 , 3615 where L(s, χ4) is the Dirichlet L-function with non-principal character mod4, γ is the Euler constant. Then after simple calculation we deduce X aN(α) πx 3 +ε τ (α)e = P (log x) + O x 4 + 3 q q 2 N(α)≤x 1 +ε 1 θ−1 + O x 2 X 2 + O xX . 1 Putting X = x 3−2θ , we obtain aN(α) πx 3 X θ0 +ε τ (α)e = P (log x) + O x + O x 4 , 3 q q 2 N(α)≤x 2−θ 1792 where θ0 = 3−2θ , θ = 3615 . 3 Because, 0.88 < θ0 < 0.89, θ0 > 4 + ε, 0 < ε < 0.01, the theorem is proved. References [1] Dadayan, Z.Yu. and A.S. Radova, Divisor function of the Gaussian integers, Visnyk Odes. Nat. Univ., 17(4) (2012), 34–39. (in Russian) 246 A. Radova and S. Varbanets [2] Gunyavy, O.A., Estimate of the integrals of type, Vistnyk Odes. Nat. Univ., phys.-math. sci., 6(3) (2001), 55–61. (in Russian) [3] Huxley, M., Area, Lattice Points and Exponential Sums, London Math. Soc. Manographs, New Series, The Clarendon Press Oxford University Press, New York, 1996, V.13, pp. 505. [4] Jutila, M., Mean value estimates for exponential sums II, Arch. Math., 55 (1991), 267–274. [5] Jutila, M., On exponential sums involving the divisor function, J. Reine Angew. Math., 355 (1985), 173–190. [6] Kr¨atzel,A., Analytische Funktionen in der Zahlentheorie, Teubner-Texte zur Mathematik, Stuttgart, 2000, V. 139, pp. 292. [7] Landau, E., Vorlesungen ¨uber Zahlentheorie, Verlag von S. Hirzel in Leipzig, 1927, pp. 308. [8] Titchmarsh, E., The Theory of the Riemann Zeta-function, Oxford, 1951, pp. 346. [9] Vandehey, J., Error term improvements for van der Corput transforms, Quart. J. Math., 65 (2014), 1461-1502. A. Radova and S. Varbanets Department of Computer Algebra and Discrete Mathematics I.I. Mechnikov Odessa National University Odessa, 65026 Ukraine radova [email protected] [email protected]