Koichi Kawada

立教大学学術リポジトリ

COMMENTARII MATHEMATICIUNIVERSITATIS SANCTI PAULIVol. 60, No. 1, 2 2011ed. RIKKYO UNIV/MATHIKEBUKURO TOKYO171–8501 JAPANNote on Sums of Seven Cubes of Smooth NumbersbyKoichi KAWADA(Received April 11, 2011)Dedicated to Professor Akio Fujii on the occasion of his official retirement1. IntroductionExtending Vinogradov’s celebrated three prime theorem, Akio Fujii [4] proved thatfor any fixed integer g ≥ 2, every sufficiently large odd integer can be written as the sumof three numbers, each of which is the product of g prime numbers. More precisely, he [4]showed that for any given positive real numbers δ1, . . . , δg satisfying δ1 + · · · + δg = 1,every sufficiently large odd integer n can be written as n = x1 + x2 + x3, where for j = 1,2 and 3, the xj takes the shapexj = pj,1pj,2 . . . pj,g with primes pj,k ≤ nδk for 1 ≤ k ≤ g .(1)In fact, he essentially established an asymptotic formula for the number of the latter repre-sentations. As Balog and Sárközy [2] pointed out, this work of Fujii [4] has relation with aproblem posed by Erdös on representations of integers by sums of numbers without largeprime factors. Such numbers are commonly called smooth numbers.The methods of Fujii [4] may be quite easily altered to show, for example, that everylarge even integer n can be written as n = 2x1 + x2 + x3 with the same constraint onxj as in (1). Thus it is immediately derived from the above work of Fujii [4] that for anyε > 0, every sufficiently large integer n, regardless of its parity, can be written as the sumof three natural numbers, each of which has no prime factor exceeding nε . Later, Balog andSárközy [2] sharpened this conclusion, replacing the last nε by exp(3√log n log log n).Further, Fujii [5] investigated additive problems of Waring’s type for smooth numbers.Amongst others, concerning sums of cubes, on which we concentrate in this short note, heestablished that every sufficiently large odd integer n can be written as n = x31 +x32 +· · ·+x39 , where each xj takes a shape similar to (1), but with the restriction pj,k ≤ nδk/3 forthe prime factors (see Theorem 3 and Corollary 3 of [5]). Again, for large even n, Fujii’sargument may lead, with trivial modifications, to the existence of the xj of the same shapesatisfying n = (2x1)3 + x32 + · · · + x39 , so essentially it follows form Corollary 3 of [5] thatfor any ε > 0, every sufficiently large n, of either parity, can be written as the sum of ninecubes of natural numbers, each of which has no prime factor exceeding nε . This conclusionwas refined by Harcos [6], who showed the assertion with exp(c√log n log log n) for somepositive absolute constant c, in place of nε . Brüdern and Wooley [3] further improvedHarcos’s result by reducing the number of summands. Namely, they [3] showed that every8990 Koichi KAWADAsufficiently large n is the sum of eight cubes of natural numbers that have no prime factorexceeding exp(c√log n log log n) with some positive absolute constant c.The purpose of this article is to discuss the corresponding problem for sums of sevencubes. Hereafter, for an integer m ≥ 2, we denote the largest prime factor of m by P(m),and we seek for an upper bound for P(x1x2 . . . x7) in terms of n, with which every suffi-ciently large integer n admits the representationn = x31 + x32 + x33 + x34 + x35 + x36 + x37 .(2)Our result here in this direction is substantially weaker than the aforementioned results,with respect to the smoothness of the variables, and we aim to show that every large n iswritten in the form (2) with P(x1 . . . x7) ≤ nσ/3 for a smaller σ . Note that Linnik’s sevencube theorem means of course that σ = 1 is admissible in the latter statement.If we would be allowed to leave one variable aside in the representation (2), then wemay make the remaining six variables satisfactorily smooth. In fact, it may be deduced froma deep theorem of Wooley ([9], Theorem 1.2) that every large n can be written in the form(2) with P(x2x3 . . . x7) ≤ exp(c√log n log log n) for some c > 0. But still considerabletechnical efforts are required to obtain a significant bound for the largest prime factor of allthe seven variables, substantially better than P(x1x2 . . . x7) ≤ n1/3. Such a bound appearedfirst in the previous work of the author [7] as a by-product of his research on sums of sevencubes of almost primes. On puttingξ =√2833 − 4341and σ0 = 1 − 4(5 − 16ξ − ξ2)8ξ + 21 ,(3)Theorem 5 of [7] asserts that for any σ > σ0, every sufficiently large n can be written inthe form (2) with natural numbers xj (1 ≤ j ≤ 7) satisfying P(x1x2 . . . x7) ≤ nσ/3. Weremark thatσ0 = (53672√2833 − 2784931)/85977is slightly smaller than 0.835239, and also that the above ξ comes from the currently bestestimate for the sixth moment of a cubic smooth Weyl sum due to Wooley ([10], Theorem1.2). In this note we reduce the lower limit for the admissible value of σ by the factor e−1/2.THEOREM. Let σ0 be defined by (3). Then for any σ > σ0e−1/2, every sufficientlylarge integer n can be written in the form (2) with natural numbers xj (1 ≤ j ≤ 7)satisfying P(x1x2 . . . x7) ≤ nσ/3.Note that σ0e−1/2 is slightly smaller than 0.506598.The Theorem is proved by adding an idea of Balog [1] to the basic work of [7]. Thisidea of Balog may be regarded as a kind of the switching principle, or the reversal roletechnique, in the sieve theory.2. Notation and the strategyThroughout the paper, we adopt the following notation and conventions.We write e(α) = exp(2πiα), and the letter p, with a subscript also, denotes primenumbers. The letter  denotes primes congruent to 2 modulo 3.Note on Sums of Seven Cubes of Smooth Numbers 91The constants ξ and σ0 are defined at (3), and σ denotes any given real number sat-isfying σ > σ0e−1/2, as in the statement of the Theorem. Then there is a positive ε suchthatσ > (σ0 + 9ε)eε−1/2,(4)and we fix such an ε with ε < 10−3 once for all, within this paper. All the implicit constants,involved in the symbols ,  and O , may depend at most on ε.With this ε, Theorem 1.2 of Wooley [10] assures the existence of a small positive ηsuch that for any U > 1, one has∫ 10∣∣∣∣∑1≤x≤2Ue(x3α)∣∣∣∣2∣∣∣∣∑1≤z≤2UP(z)≤U3ηe(z3α)∣∣∣∣4dα  U3+ξ+ε .(5)We fix an η < 10−3 having the latter property, and to facilitate the later argument, wesuppose that (2η)−1 is an integer, without loss of generality. Note that we may take this η,depending only on ε.Let n be an integer which shall be supposed to be large in terms of the ε, and putX = 12n1/3 , D = X1−σ0−9ε , M = X(14ξ−1)/(8ξ+21) .It may be understood from our minor arc estimate (see (10) below) that these choices of Dand M are optimal. And it may be helpful to record that 1 − σ0 and (14ξ − 1)/(8ξ + 21)are approximately 0.16476 and 0.10836, respectively.Next, for any U > 1, we defineA∗(U) = {l ∈ N : U1−2η < l ≤ U1−η, ∀p|l, 2U2η < p ≤ U3η} ,imitating the methods of Harcos [6] and Brüdern and Wooley [3], which shows its utilitywhen we evaluate major arc contribution. In accordance with the notation of [7], we alsodefine the setsA = {lm : l ∈ A∗(X/M), 1 ≤ lm ≤ 2X/M} ,AM = {x : M <  ≤ 2M, x ∈ A, X < x ≤ 2X} ,B = {lm : l ∈ A∗(X), X < lm ≤ 2X} .When l ∈ A∗(X/M) and 1 ≤ lm ≤ 2X/M , one has m ≤ 2(X/M)2η, while every primefactor of l exceeds 2(X/M)2η. Thus we find that every x ∈ A can be written uniquely asx = lm with l ∈ A∗(X/M). Similarly, every z ∈ B may be written uniquely as z = lmwith l ∈ A∗(X). Moreover we may notice that∀x ∈ A , P (x) ≤ (X/M)3η , and ∀z ∈ B , P (z) ≤ X3η .We also have P(y) ≤ 2M for all y ∈ AM .Then we define R1(n) to be the number of representations of n in the formn = (p1x1)3 + (p2x2)3 + y31 + y32 + y33 + z31 + z32 ,(6)92 Koichi KAWADAwhereD < pj ≤ 2D , Xpj< xj ≤ 2Xpj, yk ∈ AM , zj ∈ B ,(7)for j = 1, 2, and k = 1, 2, 3, and in addition, P(x1) ≤ Xσ . We may provide an asymptoticformula for R1(n), being based on the minor arc estimate contained essentially in Lemma5 of [7], as we shall show indeed in the sequel. Then by the fact that R1(n) > 0 for everylarge n, one obtains a conclusion which is tantamount to Theorem 5 of [7] referred after(3).To prove the Theorem, we estimate the number, say R2(n), of the representations of nin the form (6) where the variables satisfy (7) and additional conditions P(x1) ≤ Xσ andP(x2) > Xσ . Further we write R3(n) for the number of the representations of n in the form(6) where the variables satisfy (7) and P(x2) > Xσ .Now note first that trivially one has R2(n) < R3(n). So if we could show thatR1(n) > R3(n) ,(8)then we see that R1(n) > R2(n), which obviously means that n can be written in the form(6) with the variables satisfying (7), P(x1) ≤ Xσ and P(x2) ≤ Xσ . Thus the Theoremfollows from (8) at once.We remark that in the above strategy, some additional minor efforts allow us to replacethe variables pj , yk and zj for j = 1, 2 and k = 1, 2, 3, by suitable numbers having noprime factor exceeding exp(c√log n log log n) with some c > 0, although such modifica-tions have no impact on the statement of the Theorem. Here we adopt the above setting forthe ease of reference to [7].3. The circle method and minor arc estimatesIn order to show (8) by the circle method, we introduce several exponential sums.When d is a natural number and C is a finite set of integers, we writef (α; d) =∑X/d<x≤2X/de(d3x3α) , g(α; C) =∑x∈Ce(x3α) ,f1(α; d) =∑X/d<x≤2X/dP (x)≤Xσe(d3x3α) , f3(α; d) =∑X/d<x≤2X/dP (x)>Xσe(d3x3α) .For ν = 1 and 3, we also putFν(α) =∑D<p≤2Dfν(α; p) ,and then observe, by the orthogonality, thatRν(n) =∫ 10∑D<p≤2Df (α; p)Fν(α)g(α;AM)3g(α;B)2e(−nα)dα .Note on Sums of Seven Cubes of Smooth Numbers 93To evaluate the last integral, we adopt the same Hardy-Littlewood dissection as in [7].Let P be the set of real numbers α ∈ [0, 1) such that there exist coprime integers q and asatisfying|α − a/q| ≤ (log X)500X−3 , 1 ≤ q ≤ (log X)500 , 0 ≤ a ≤ q ,(9)and write p = [0, 1) \ P. Then the contribution of the minor arcs p to the integral repre-senting Rν(n) may be swiftly handled by reference to the upper bound for E contained inLemma 5 of [7]. In fact, on writingI =∫ 10|g(α;A)|6dα and Jν =∫ 10|Fν(α)2g(α;B)4|dα ,for ν = 1 and 3, and on checking the requirementmax{X1/10,X1/9D−1/3} < M ≤ (X/D)1/7 ,we may obtain the estimate∑D<d≤2D∣∣∣∣∫pf (α; d)Fν(α)g(α;AM)3g(α;B)2e(−nα)dα∣∣∣∣ X3/4+εM3/4D1/4J 1/2ν (X3/2 + (MI)1/2 + (XI)1/3M3/2D1/4)+ X4(log X)−50 ,(10)by (4.20) of [7] (see also (4.27)–(4.29), (4.39), (4.40) and (4.42) of [7]). Here only onecomment may be in order; on this derivation, we make Fν(α) play the role of g(α;B1) in[7], and this change has an actual effect only on estimating T2 which should readT2 =∫ 10|g(α;AM)2Fν(α)2g(α;B)4|dα ,in the current context. But, by orthogonality, T2 is equal to the number of solutions of theequationy31 + (p1x1)3 + z31 + z32 = y32 + (p2x2)3 + z33 + z34 ,subject to yj ∈ AM , D < pj ≤ 2D, X/pj < xj ≤ 2X/pj and zk ∈ B for j = 1, 2 and1 ≤ k ≤ 4. Since every natural number less than 2X has at most O(1) prime divisors inthe interval (D, 2D], we still haveT2 ∫ 10|g(α; N ∩ [1, 2X])|8dα  X5 ,as is claimed in [7], by virtue of a theorem of Vaughan. Thus we may confirm the estimate(10).Then, by (5), together with the above remark concerning Fν(α), we have I (X/M)3+ξ+ε and Jν  X3+ξ+ε , for ν = 1 and 3. Consequently, a modicum of straight-forward computation may deduce from (10) that∫p∑D<p≤2Df (α; p)Fν(α)g(α;AM)3g(α;B)2e(−nα)dα94 Koichi KAWADA X4+2ε(X− 5−16ξ−ξ28ξ+21 D1/4 + X−2(5−16ξ−ξ2)8ξ+21 D1/2)+ X4(log X)−50 ,whence we thus far conclude, for ν = 1 and 3, thatRν(n) =∫P∑D<p≤2Df (α; p)Fν(α)g(α;AM)3g(α;B)2e(−nα)dα+ O(X4(log X)−50) .(11)It is essentially a routine to evaluate the contribution of the major arcs P, and weexecute it in the next section.4. Contribution of the major arcsWe begin with a simple lemma concerning the set A∗(U).LEMMA. Under our convention on η (see the comment after (5)), we have∑l∈A∗(U) l−1  1, provided that U is sufficiently large in terms of η (so ultimately, interms of ε).Proof. Recall that (2η)−1 = h, say, is an integer. Then, whenever p1, . . . , ph−1 areprimes in the interval (2U2η, U2η+2η2], one hasU1−2η < (2U2η)h−1 < p1p2 . . . ph−1 ≤ (U2η+2η2)h−1 < U1−η,so that p1p2 . . . ph−1 ∈ A∗(U). And it is trivial that each natural number can be written asthe product of (h − 1) primes in at most (h − 1)! ways. These facts reveal that∑l∈A∗(U)l−1 ≥ 1(h − 1)!( ∑2U2η<p≤U2η+2η2p−1)h−1,and the sum over p on the right hand side is log(1 + η) + O((log U)−1), by Mertens’theorem. Thus the Lemma follows immediately. For integers q and a with q ≥ 1, and a real number β, we defineS(q, a) =q∑x=1e(ax3/q) and v(β) =∫ 2XXe(t3β)dt .For any q and a with (q, a) = 1, and for any β, we know thatS(q, a)  q2/3 and v(β)  X(1 + X3|β|)−1 .(12)In fact, the former is Theorem 4.2 of Vaughan [8], and the latter follows by the partialintegration with a trivial bound.Note on Sums of Seven Cubes of Smooth Numbers 95Now let α ∈ P, and q and a be coprime integers satisfying (9), and write β = α−a/q .Then, for any integer d , we deduce from Theorem 4.1 of Vaughan [8] thatf (α; d) = q−1S(q, ad3)∫ 2X/dX/de(d3t3β)dt + O((log X)501) .Noting further that S(q, ad3) = S(q, a) whenever d is coprime to q , we find in the lattercircumstances thatf (α; d) = (qd)−1S(q, a)v(β) + O((log X)501) .(13)Accordingly we have the formulae∑D<p≤2Df (α; p) = ∆1q−1S(q, a)v(β) + O(D(log X)501) ,(14)g(α;Am) =∑M<≤2M∑l∈A∗(X/M)f (α; l)= ∆2q−1S(q, a)v(β) + O(M( XM)1−η(log X)501),(15)g(α;B) =∑l∈A∗(X)f (α; l)= ∆3q−1S(q, a)v(β) + O(X1−η(log X)501) ,(16)where∆1 =∑D<p≤2D1p, ∆2 =∑M<≤2M1∑l∈A∗(X/M)1l, ∆3 =∑l∈A∗(X)1l.As for these quantities, the lower bounds∆1  (log X)−1 , ∆2  (log X)−1 and ∆3  1(17)may be confirmed by our Lemma, Mertens’ theorem and its variant for arithmetic progres-sions. As regards the upper bounds, our argument needs nothing more than the trivial oneslike ∆j  log X for j = 1, 2 and 3.Since the measure of P is O((log X)1500X−3), and all the error terms appearing in theformulae (14)-(16) are O(X1−η/2), it follows simply from these formulae that∫P∑D<p≤2Df (α; p)Fν(α)g(α;AM)3g(α;B)2e(−nα)dα= ∆1∆32∆23∑q≤(log X)500q∑a=1(a,q)=1V6(q, a, n)Wν(q, a, n) + O(X4−η/3) ,(18)where, for s ∈ N and ν = 1 and 3, we writeVs(q, a, n) = (q−1S(q, a))se(−an/q) ,96 Koichi KAWADAandWν(q, a, n) =∫ (log X)500X−3−(log X)500X−3Fν(aq+ β)v(β)6e(−nβ)dβ .We next look into F3(α). Noting that Xσ >√2X/D, we havef3(α; p) =∑p1>Xσf (α; pp1) ,for any p with D < p ≤ 2D. The last sum is actually finite, because f (α; pp1) vanishesfor p1 > 2X/D. And, on putting X1 = XD−1(log X)−510, the contribution of the primesp1 > X1 is∑X1<p1≤2X/DXpp1 Xp(log X)−1 log log X ,by Mertens’ theorem. Hence we haveF3(α) =∑D<p≤2D∑Xσ <p1≤X1f (α; pp1) + O( ∑D<p≤2DX log log Xp log X),and therefore, by (13), when q and a are coprime integers satisfying (9), we haveF3(α) = ∆1Ξq−1S(q, a)v(β) + O(∆1X(log X)−1 log log X) ,(19)where β = α − a/q andΞ =∑Xσ <p1≤X11p 1= log(log(X/D)σ log X)+ O(log log Xlog X),by Mertens’ theorem again. And, by (4), we see thatΞ = log(σ0 + 9εσ)+ O(log log Xlog X)<1 − ε2.(20)After we get the formula (19) on F3(α), what remains to execute is completely routineprocedure. We have V6(q, a, n)  q−2 by the former bound in (12), so applying (19) wesee∑q≤(log X)500q∑a=1(a,q)=1V6(q, a, n)W3(q, a, n)= ∆1Ξ∑q≤(log X)500q∑a=1(a,q)=1V7(q, a, n)∫ (log X)500X−3−(log X)500X−3v(β)7e(−nβ)dβ+ O( ∑q≤(log X)500q−1∆1Xlog log Xlog X∫ ∞−∞|v(β)|6dβ),Note on Sums of Seven Cubes of Smooth Numbers 97and the last error is O(∆1X4(log X)−1(log log X)2), by the latter bound in (12). Now weintroduce the familiar singular series and the singular integral associated with sums of sevencubes, defined respectively byS(n) =∞∑q=1q∑a=1(a,q)=1V7(q, a, n) , I (n) =∫ ∞−∞v(β)7e(−nβ)dβ .It follows quite easily from (12) that∫ (log X)500X−3−(log X)500X−3v(β)7e(−nβ)dβ = I (n) + O(X4(log X)−3000) ,and also that∑q≤(log X)500q∑a=1(a,q)=1V7(q, a, n) = S(n) + O((log X)−500/3) .Moreover, one obtains the familiar bounds1  S(n)  1 , X4  I (n)  X4 .(21)Actually, the both upper bounds are immmediate by (12). The lower bound for S(n) iscontained in Theorem 4.6 of Vaughan [8]. As for the lower bound for I (n), one appliesFourier’s inversion formula, and gets the expressionI (n) =∫· · ·∫137(u1u2 . . . u6(n − u1 − u2 − · · · − u6))−2/3du1 . . . du6 ,where the domain of the integral is defined by X3 ≤ uj ≤ (2X)3 (1 ≤ j ≤ 6) andX3 ≤ n − u1 − u2 − · · · − u6 ≤ (2X)3. Then, recalling that (2X)3 = n and consideringthe contribution of the domain defined by X3 ≤ uj ≤ (7/6)X3 (1 ≤ j ≤ 6), for example,one may confirm the claimed lower bound for I (n).We gather these results to compute the right hand side of (18) for ν = 3, and then,recalling (11) with (17) and (21), we conclude thatR3(n) = ∆21∆32∆23S(n)I (n)(Ξ + O((log X)−1(log log X)2)) .(22)We next turn to R1(n). From (14), (19) and the trivial formula f1(α; p) = f (α; p) −f3(α; p), we may deduce a formula for F1(α) corresponding to (19), with 1 − Ξ in placeof Ξ . Then, through the same lines as the case of R3(n) above, we obtain a formula forR1(n) similar to (22), in which Ξ is replaced by 1 − Ξ . Hence we haveR1(n) − R3(n) = ∆21∆32∆23S(n)I (n)(1 − 2Ξ + O((log X)−1(log log X)2)) .In view of (20), therefore, the desired inequlity (8) is established, provided that n is suffi-ciently large in terms of ε. This completes the proof of the Theorem, as we already observedin §2.98 Koichi KAWADAReferences[ 1 ] A. Balog, On additive representation of integers, Acta Math. Hungar. 54 (1989), 297–301.[ 2 ] A. Balog and A. Sárközy, On sums of integers having small prime factors, I, Studia Sci. Math. Hungar.19 (1984), 35–47.[ 3 ] J. Brüdern and T. D. Wooley, On Waring’s problem for cubes and smooth Weyl sums, Proc. London Math.Soc. (3) 82 (2001), 89–109.[ 4 ] A. Fujii, An additive problem in theory of numbers, Acta Arith. 40 (1981), 41–49.[ 5 ] A. Fujii, Some additive problems of numbers, in “Elementary and analytic theory of numbers” (Warsaw,1982), 121–141, Banach Center Publ., 17, PWN, Warsaw, 1985.[ 6 ] G. Harcos, Waring’s problem with small prime factors, Acta Arith. 80 (1997), 165–185.[ 7 ] K. Kawada, On sums of seven cubes of almost primes, Acta Arith. 117 (2005), 213–245.[ 8 ] R. C. Vaughan, The Hardy-Littlewood Method, 2nd ed. Cambridge Univ. Press, 1997.[ 9 ] T. D. Wooley, Breaking classical convexity in Waring’s problem: Sums of cubes and quasi-diagonalbehaviour, Invent. Math. 122 (1995), 421–451.[ 10 ] T. D. Wooley, Sums of three cubes, Mathematika 47 (2000), 53–61.Department of MathematicsFaculty of EducationIwate UniversityMorioka, 020–8550 Japane-mail: kawada@iwate-u.ac.jp

Note on Sums of Seven Cubes of Smooth Numbers

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Note on Sums of Seven Cubes of Smooth Numbers

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