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Incomplete Kloosterman Sums to Prime Power Modules

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Incomplete Kloosterman Sums to Prime Power Modules Bulletin T. CLIV de l’Acade´mie serbe des sciences et des arts 2021 − Classe des Sciences mathe´matiques et naturelles Sciences mathe´matiques, o 46 N INCOMPLETE KLOOSTERMAN SUMS TO PRIME POWER MODULES M.A˙ . KOROLEV, I. S. REZVYAKOVA To the Blessed Memory of Professor Aleksandar Pavle Ivic´ (6.3.1949 – 27.12.2020) (Presented at the 3nd Meeting, held on April 23, 2021) A b s t r a c t. We prove that for prime p, p + , integer r 4 and q = pr an → ∞ ! incomplete Kloosterman sum of length N to modulus q can be estimated non-trivially (with 1/(r 1) power-saving factor) for very small N, namely, for N (q log q) − . ≫ AMS Mathematics Subject Classification (2020): 11L05. Key Words: Kloosterman sums, inverse residues, Karatsuba’s method, powerful mod- ules. 1. Introduction Let q, a, b be integers, q ! 3, (a, q) = 1, and let 1 < N < N1 < q, N1 " 2N. An incomplete Kloosterman sum is an exponential sum of the type ′ Sq(N) = Sq(a, b; N, N1) = eq(an + bn). N<!n!N1 2πiα Here, as usual, eq(u) = e(u/q), e(α) = e , prime sign means the summation over n coprime to q and n = 1/n denotes the inverse residue to n modulo q, that is, the 74 M.A˙ . Korolev, I. S. Rezvyakova solution of the congruence nn 1 (mod q). Both the classical A. Weil’s bound ≡ p 1 − ep(an + bn) " 2√p " n=1 " " ! " " " for prime p (see [1] and [2, A"ppendix V, 11]) a"nd “multiplicative” property of com- plete Kloosterman sums q ′ Sq(a, b) = eq(an + bn) n=1 ! r together with the precise expressions for Sq(a, b), q = p , r ! 2 (see, for example, [3, Lemma 4b], [4, formulas (50)]) lead to the estimate S (N) √q τ(q) log q, | q | ≪ where τ(q) denotes divisor function. Since τ(q) qε for any fixed ε > 0, this ≪ estimate in non-trivial for N q0.5+ε. ≫ In the case N " √q, the problem of estimating the sum Sq(N) was open until the early 1990s, when A.A. Karatsuba invented a new method (see [5]–[8]) that allows one to get the bound of the type N S (N) . | q | ≪ (log q)c In the case of an arbitrary modulus q, this bound holds for any N qε with some c, ≫ 0 < c < 0.5. For prime q = p, Karatsuba’s method allows one to obtain a non-trivial 2/3+ε bound of S (N) for very small N, namely, for N e(log q) (see [9], [10])1. q ≫ The estimate of Sq(N) for N " √q in general case with power-saving factor is still open problem. However, such estimate is possible for some special modules, namely, for q = pr, where p is prime and r is large integer. In such case, A.A. Karat- suba noted [6, p. 722] that the identity 1 2 2 r 1 r 1 r 1 1 px + p x + ( 1) − p − x − (mod q) 1 + px ≡ − − · · · − allows one to reduce the incomplete Kloosterman sums to the exponential sums with polynomials. These sums can be treated by I.M. Vinogradov’s method2. The im- plementation of such idea allows to the first author [14, Theorem 4] to prove the following theorem: 2/3+ε 1A.A. Karatsuba himself expected to obtain non-trivial estimates for the case N e(log q) ; ≫ see the facsimile of his manuscript dated by October 19, 2007 on page 356 of [11]. 2The idea of such reduction to the exponential sums with polynomials belongs to A.G. Postnikov who applied I.M. Vinogradov’s method to short character sums. For the details, see [12], [13]. Incomplete Kloosterman sums to prime power modules 75 r Let p ! 2 be a fixed prime, (a, p) = 1, r ! 72, q = p , and suppose that 24 1/3 p " N " q . Then, for any 1 " µ " q, the sum ′ S = eq(an + bn) µ<n!!µ+N obeys the estimate 3 c(log N) 4 S 6N exp , c = 13− | | " − (log q)2 # $ (see also [15]). One can check that this estimate is non-trivial for very small N, namely, for 2/3 N ! exp c1(log q) , and provides the power saving decreasin%g factor for N& qα with any fixed α: ≍ 1 δ 3 S N − , δ α . | | ≪ ≍ The first author also obtained a generalization of this theorem to the case of so-called “powerful” modules, that is, to the modules q such that the radical d(q) = p is p q | much smaller that q at logarithmic scale: ' 4 Suppose that q is sufficiently large integer, c1 = 900, c = 160− , and suppose that 15 c (log q)2/3 max d , e 1 " N " √q. Then for any a, b, (a, q) = 1 w%e have: & c(log N)3 S N exp . | | " − (log q)2 # $ Thus, the case q = p is the most complicated for studying, and the case of “very powerful” modulus q = pr, r + , is the easiest. The purpose of the present paper → ∞ is to study the “middle case” q = pr, where r is a fixed integer and prime p + , → ∞ and to find out when the non-trivial bound for Sq(N), N " √q, is available. It appears that it is possible to do for all r ! 4. Namely, the following assertion holds true. r Theorem 1.1. Suppose that r ! 4 is fixed, p is prime and q = p . Further, let 1 r 1 3/5 q log q − " N " q . % & 76 M.A˙ . Korolev, I. S. Rezvyakova Then the following inequality holds: q log q γr Sq(N) " cN , | | N r 1 # − $ where c = c(r) > 0 and γ 1 = r2(r 1)2 + (4r + 1)(r 1) + 4. r− − − Remark 1.1. One can check that 1 1 1 1 γ4− = 199, γ5− = 488, γ6− = 1029, γ7− = 1942, 1 1 1 γ8− = 3371, γ9− = 5484, γ1−0 = 8473. 1 +ε r 1 1 In the case N ! q − , it is possible to prove a slightly better estimate with γr− = (r 1)2(r 2)2 + r 1. To do this, one should take k = 1 r(r 1) instead of − − − 2 − k = 1 r(r 1) + 1 in the proof of theorem. If 2 − 1 +ε 1 + 1 r 1 r 1 r(r 1) q − " N " q − − , then the application of H. Weyl’s method yields to the similar estimate with γr = 2 r 2 − . Such estimate is more precise than the previous for 4 " r " 18. 3/5 If q < N " q then the results from [1]–[4] imply much more precise estimate 2 3 Sq(N) √q τ(q) log q. The same bound holds true in the cases q = p , p and | | ≪ 2 3 √q log q N " q. The problem of estimating Sq(N) for q = p , p and N " √q ≪ 3 is still open. However, in the case q = p , we obtain an estimate for Sq(N) which is slightly better than the trivial. Namely, the following assertion is true. 3 2/3 Theorem 1.2. Suppose that q = p and √q N 0.5q . Then Sq(N) ≪ " | | ≪ √q, where the implied constants are absolute. 2. Auxilliary assertions The proof of the above Theorem 1.1 is quite simple but the assertion seems to be new. It relies on the upper bound for the integral Jk,s(P ) of I. M. Vinogradov. Suppose that k, s ! 2 are integers, P ! 2. Then Jk,s(P ) coincides with the number of integral solutions of the system x + + x = x + + x , 1 · · · k k+1 · · · 2k x2 + + x2 = x2 + + x2 , ( 1 · · · k k+1 · · · 2k ) . ) . *) s s s s x1 + + xk = xk+1 + + x2k, ) · · · · · · ) with 1 " x1, . , x2k +)" P . The following inequality holds true. Incomplete Kloosterman sums to prime power modules 77 Lemma 2.1. For any fixed ε > 0, for any fixed k, s ! 2 there exists the constant c = c(ε, k, s) > 0 such that the inequality 1 ε k 2k s(s+1) Jk,s(P ) " cP P + P − 2 . , - 1 ε holds for P ! 2. Moreover, if k > 2 s(s + 1) then the factor P can be dropped and the constant c will depend on k, s only. This estimate was proved partially by T. Wooley and then by J. Bourgain, C. Demeter, L. Guth for an arbitrary k, s ! 2 (see [16] for the proof and [17] for short survey and references). We also need the following simple lemma. Lemma 2.2. Let p ! 3 be a prime number, (a, p) = 1, and suppose that Λ is integer, 1 < Λ < p/2. Then the sum Λ aµλ V = e p µ=1" λ <Λ " !" |!| , -" " " obeys the estimate V < 2p log p. " " PROOF. For any µ under considering, we have (µ, p) = 1. Hence, Λ 1 1 1 aµλ − aµλ πaµ − aµ − e 1 + 2 e 1 + 2 sin 1 + , p " p " p " p " λ <Λ " " λ=1 " " " . " |!| , -" " ! , -" " " . " " " " " " . wh"ere, as usual, ξ" means "the distance be"tween ξ a"nd closes"t integer. S.ince.all the ) ) numbers aµ, 1 " µ " Λ are pairwisely noncongruent modulo p and Λ < p/2 then Λ 1 1 − aµ − p 1 p V 1 + Λ + p + p log + 1 , " p " x " 2 2 µ=1# . $ 1 x (p 1)/2 # $ ! . ! !!− . i.e., V < 2p log p. Lemma 2.3. There exists an absolute constant c0 > 0 such that for any integers q, a, b and N satisfying the conditions q ! 2, (a, q) = 1, 1 < N < q, the following inequality holds: N 2 eq(ax + bx) " c0"q.
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