We apply recent progress on Vinogradov's mean value theorem to improve bounds
for the function $H(k)$ in the Waring-Goldbach problem. We obtain new results
for all exponents $k \ge 7$, and in particular establish that for large $k$ one
has \[H(k)\le (4k-2)\log k-(2\log 2-1)k-3.\

Kumchev, Angel

Wooley, Trevor D

arXiv

We apply recent progress on Vinogradov's mean value theorem to improve bounds for the function H(k) in the Waring-Goldbach problem. We obtain new results for all exponents k≥7, and in particular establish that for large k one has H(k)≤(4k−2)logk−(2log2−1)k−3.</b

Explore Bristol Research

Kumchev, A., & Wooley, T. D. (2017). On the Waring–Goldbachproblem for seventh and higher powers. Monatshefte für Mathematik,183(2), 303–310. https://doi.org/10.1007/s00605-016-0936-7Peer reviewed versionLink to published version (if available):10.1007/s00605-016-0936-7Link to publication record on the Bristol Research PortalPDF-documentThis is the accepted author manuscript (AAM). The final published version (version of record) is available onlinevia Springer Verlag at http://doi.org/10.1007/s00605-016-0936-7. Please refer to any applicable terms of use ofthe publisher.University of Bristol – Bristol Research PortalGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/brp-terms/ON THE WARING–GOLDBACH PROBLEMFOR SEVENTH AND HIGHER POWERSANGEL V. KUMCHEV AND TREVOR D. WOOLEYAbstract. We apply recent progress on Vinogradov’s mean value theorem to improvebounds for the function H(k) in the Waring–Goldbach problem. We obtain new results forall exponents k ≥ 7, and in particular establish that for large k one hasH(k) ≤ (4k − 2) log k − (2 log 2− 1)k − 3.1. IntroductionIn our recent work [7], we reported on the consequences for the Waring-Goldbach problemof recent progress on Vinogradov’s mean value theorem based on efficient congruencing(see, for example, [9, 10]). We now revisit our analysis in order to incorporate the latestdevelopments stemming from work of Bourgain, Demeter and Guth [1]. We first recall thedefinition of the function H(k) associated with the Waring-Goldbach problem. Consider anatural number k and prime number p, and define θ = θ(k, p) to be the integer with pθ|kbut pθ+1 - k, and γ = γ(k, p) byγ(k, p) ={θ + 2, when p = 2 and θ > 0,θ + 1, otherwise.We then put K(k) =∏(p−1)|k pγ, and denote by H(k) the least integer s such that everysufficiently large positive integer congruent to s modulo K(k) may be written as the sum ofs k-th powers of prime numbers.Improving on the bound H(k) ≤ k(4 log k + 2 log log k + O(1)), as k → ∞, due to Hua[3, 4], we recently showed that H(k) ≤ (4k − 2) log k + k − 7. The improved bound that wenow present in this note saves roughly (2 log 2)k further variables.Theorem 1. When k is large, one has H(k) ≤ (4k − 2) log k − (2 log 2− 1)k − 3.For small values of k one has the boundsH(1) ≤ 3, H(2) ≤ 5, H(3) ≤ 9, H(4) ≤ 13, H(5) ≤ 21, H(6) ≤ 32, H(7) ≤ 46,as a consequence of work of Vinogradov [8], Hua [2], Kawada and the second author [5], thefirst author [6], and Zhao [11]. For larger values of k, we recently established thatH(8) ≤ 61, H(9) ≤ 75, H(10) ≤ 89, H(11) ≤ 103, H(12) ≤ 117,H(13) ≤ 131, H(14) ≤ 147, H(15) ≤ 163, H(16) ≤ 178,H(17) ≤ 194, H(18) ≤ 211, H(19) ≤ 227, H(20) ≤ 244.We now obtain the following bounds for H(k) when 7 ≤ k ≤ 20.2000 Mathematics Subject Classification. 11P32, 11L20, 11P05, 11P55.1Theorem 2. Let 7 ≤ k ≤ 20. Then H(k) ≤ s(k), where s(k) is defined by Table 1.k 7 8 9 10 11 12 13 14 15 16 17 18 19 20s(k) 45 57 69 81 93 107 121 134 149 163 177 193 207 223Table 1. Upper bounds for H(k) when 7 ≤ k ≤ 20Our proof of Theorems 1 and 2 proceeds by directly incorporating the refinements availablevia [1] into our previous methods from [7]. We record improved estimates for Weyl sums in§2, both pointwise bounds and mean value estimates. Then, in §3, we indicate how to refineour previous bounds for H(k) using these bounds, thereby establishing Theorems 1 and 2.Throughout this paper, the letter ε denotes a sufficiently small positive number. Wheneverε occurs in a statement, we assert that the statement holds for each positive ε, and anyimplied constant in such a statement is allowed to depend on ε. The letter p, with orwithout subscripts, is reserved for prime numbers. We also write e(x) for exp(2πix), and(a, b) for the greatest common divisor of a and b. Finally, for real numbers θ, we denote bybθc the largest integer not exceeding θ, and by dθe the least integer no smaller than θ.2. Auxiliary estimates for exponential sumsWe refine the work of [7, §§2 and 3] by incorporating recent progress on Vinogradov’smean value theorem due to Bourgain, Demeter and Guth [1]. Recall the classical Weyl sumfk(α;X) =∑X<x≤2Xe(αxk),in which we suppose that k ≥ 2 is an integer and α is real. When k ≥ 3 is an integer, wedefine σk by means of the relation(2.1) σ−1k = min{2k−1, k(k − 1)}.Also, for k ≥ 3, we define the multiplicative function wk(q) by takingwk(puk+v) ={kp−u−1/2, when u ≥ 0 and v = 1,p−u−1, when u ≥ 0 and 2 ≤ v ≤ k.Lemma 2.1. Suppose that k ≥ 3. Then either one has fk(α;X) X1−σk+ε, or there existintegers a and q such that 1 ≤ q ≤ Xkσk , (a, q) = 1 and |qα− a| ≤ X−k+kσk , in which casefk(α;X)wk(q)X1 +Xk|α− a/q|+X1/2+ε.Proof. One may apply the argument of the proof of [7, Lemma 2.1], noting only that the re-finement of [10, Theorem 11.1] that follows by employing the bounds recorded in [1, Theorem1.1] permits the use of the exponent σk with the revised definition (2.1) presented above. We also require upper bounds for the corresponding Weyl sum over prime numbers,gk(α;X) =∑X<p≤2Xe(αpk),and these we summarise in the next lemma.2Lemma 2.2. Suppose that k ≥ 4 and X2σk/3 ≤ P ≤ X9/20. Then either one has the boundgk(α;X)  X1−σk/3+ε, or else there exist integers a and q such that 1 ≤ q ≤ P , (a, q) = 1and |qα− a| ≤ PX−k, in which case(2.2) gk(α;X)X1+ε(q +Xk|qα− a|)1/2.Proof. One may follow the argument of the proof of [7, Lemma 2.2], noting that the refine-ment to the exponent σk made available via (2.1) as exhibited in Lemma 2.1. In order to describe our critical mean-value estimate, we introduce a set of admissibleexponents for kth powers as follows. Let t = tk and u = uk be positive integers to be fixedin due course. Put θ = 1− 1/k, and define(2.3) λi = (θ + σk−1/k)i−1 (1 ≤ i ≤ u+ 1).Then define λu+2, . . . , λu+t by puttingλu+2 =k2 − θt−3k2 + k − kθt−3λu+1,(2.4)λu+j =k2 − k − 1k2 + k − kθt−3θj−3λu+1 (3 ≤ j ≤ t),(2.5)and then write(2.6) Λ = λ1 + . . .+ λt+u.Lemma 2.3. Let k, t and u be positive integers with k ≥ 3 and t ≥⌊12(k + 3)⌋, and let wbe a non-negative integer. Define the exponents λj and Λ by means of (2.3)-(2.6), and putη = max{0, k − Λ− 2wσk}. Then when N is sufficiently large, one has∫ 10|gk(α;N)|2wt+u∏j=1∣∣gk(α;Nλj)∣∣2 dα N2Λ+2w−k+η+ε.Proof. This is [7, Lemma 3.3], modified to reflect the improved Weyl exponent (2.1) asexhibited in Lemmata 2.1 and 2.2. 3. The upper bound for H(k)An upper bound for H(k) follows by combining the mean value estimate supplied byLemma 2.3 with the Weyl-type estimate stemming from Lemma 2.2.Lemma 3.1. Let k, t and u be positive integers with k ≥ 3 and t ≥⌊12(k + 3)⌋. Define theexponent Λ by means of (2.6), and put v = b(k−Λ)/(2σk)c and η∗ = k−Λ− 2vσk. Finally,defineh =1, when 0 ≤ η∗ < 12σk,2, when 12σk ≤ η∗ < σk,3, when σk ≤ η∗ < 2σk.Suppose in addition that 2(t + u + v) + h ≥ 3k + 1 and, when h ∈ {1, 2}, that either v ≥ 3or η∗ < hσk/3. ThenH(k) ≤ 2(t+ u+ v) + h.3Proof. This is [7, Lemma 4.1], modified to reflect the improved Weyl exponent (2.1) asexhibited in Lemma 2.1-2.3. We establish Theorems 1 and 2 by applying Lemma 3.1. Recall (2.3)-(2.6), and writeσ = σk−1 and φ = θ + σ/k. Then, just as in the discussion of [7, §5], one hask − Λ = − kσ1− σ+(k2(k + 1)σ + θt−3((k3 − 3k2 + k + 2)− σ(k3 − 2k2 + k + 2))(k2 + k − kθt−3)(1− σ))φu.The proof of Theorem 2. Let k be an integer with 7 ≤ k ≤ 20, and define t = tk, u = uk,v = vk and h = hk by means of Table 2. Then by application of a simple computer program,one confirms the validity of the hypotheses of Lemma 3.1. Thus H(k) ≤ 2(t + u + v) + h.Indeed, with h∗k defined as in Table 3, one finds that for each k one has 2η∗/σk < h∗k. Wenote in this context that the entries in this table have been rounded up in the final decimalplace presented. This completes our proof of Theorem 2.k 7 8 9 10 11 12 13 14 15 16 17 18 19 20tk 7 12 17 10 13 10 24 19 30 17 25 18 29 37uk 13 12 14 25 29 37 28 42 41 60 56 74 66 63vk 2 4 3 5 4 6 8 5 3 4 7 4 8 11hk 1 1 1 1 1 1 1 2 1 1 1 1 1 1Table 2. The values of tk, uk, vk and hk for 7 ≤ k ≤ 20k 7 8 9 10 11 12 13h∗k 0.44643 0.22927 0.02678 0.00739 0.97975 0.00042 0.08628k 14 15 16 17 18 19 20h∗k 1.94435 0.03925 0.01091 0.39085 0.00541 0.52855 0.00043Table 3. The values of h∗k for 7 ≤ k ≤ 20As we remarked in [7, §5], the non-monotonicity in the values of tk, uk and vk recorded inTable 2 is a consequence of the fact that θ and φ are close in size, and thus the optimisationis sensitive only to the sum tk + uk rather than the individual values of tk and uk.The proof of Theorem 1. We adapt the proof of [7, Theorem 1], supposing throughout thatk is sufficiently large. Put t = tk and u = uk, wheretk =⌈12k log k⌉and uk = dk(2 log k − log 2)e − t.It is convenient to define γ = dk(2 log k − log 2)e − k(2 log k − log 2). Also, we writeτ =1k(k − 1)and σ =1(k − 1)(k − 2),4so that σk = τ and σk−1 = σ. Our earlier formula for k − Λ now takes the shapek − Λ = − kσ1− σ+(k2(k + 1)(k − 1)3σ + θtk3(k3 − 3k2 +O(k))(k − 1)3(k2 + k − kθt−3)(1− σ))φu.As in the corresponding proof of [7, Theorem 1], one finds thatθt = e−t/k(1− log k4k+O(k−3/2)) k−1/2.Also, sincelog φ = log(1− 1− σk)= −1k− 12k2+O(1k3),one discerns thatφu = e−u/k(1− 3 log k − 2 log 24k+O(k−3/2)) k−3/2.Consequently,k − Λ = − kσ1− σ+(k − 1 +O(k−1/2))θtφu +O(k−5/2),whereθtφu = e−(t+u)/k(1− 2 log k − log 22k+O(k−3/2))= e−γ/k(2k2− 2 log k − log 2k3+O(k−7/2)).Sinceστ=k(k − 1)(k − 1)(k − 2)= 1 +2k+O(1k2),we find thatk − Λ2τ= −12(k + 2) +e−γ/kk(k − 1)k3(k − 1)(k − log k + 12log 2) +O(k−1/2)= −12(k + 2) + (k − log k + 12log 2− 2) (1− γ/k) +O(k−1/2)= 12k − log k − 3 + 12log 2− γ +O(k−1/2).Put v = b(k − Λ)/(2τ)c, set η∗ = k − Λ − 2vτ , and define h as in the statement of Lemma3.1. Then one has 0 ≤ η∗ < 2τ , and in all circumstances one may confirm that2v + h =k − Λ− η∗τ+ h ≤ k − Λτ+ 2 ≤ k − 2 log k − 4 + log 2− 2γ +O(k−1/2).Since2(t+ u+ v) + h ≤ 2(2k log k − k log 2 + γ) + k − 2 log k − 4 + log 2− 2γ +O(k−1/2),we therefore conclude from Lemma 3.1 thatH(k) ≤ (4k − 2) log k − (2 log 2− 1)k − 4 + log 2 +O(k−1/2).We have assumed k to be sufficiently large, and thus we have established the boundH(k) ≤ (4k − 2) log k − (2 log 2− 1)k − 3.5This completes the proof of Theorem 1. References[1] J. Bourgain, C. Demeter and L. Guth, Proof of the main conjecture in Vinogradov’s mean value theoremfor degrees higher than three, preprint available as arXiv:1512.01565.[2] L. K. Hua, Some results in prime number theory, Quart. J. Math. Oxford Ser. 9 (1938), 68–80.[3] L. K. Hua, Additive Primzahltheorie, B. G. Teubner Verlagsgesellschaft, Leipzig, 1959.[4] L. K. Hua, Additive Theory of Prime Numbers, American Mathematical Society, Providence, RI, 1965.[5] K. Kawada and T. D. Wooley, On the Waring–Goldbach problem for fourth and fifth powers, Proc.London Math. Soc. (3) 83 (2001), no. 1, 1–50.[6] A. Kumchev, The Waring–Goldbach problem for seventh powers, Proc. Amer. Math. Soc. 133 (2005),no. 10, 2927-2937.[7] A. V. Kumchev and T. D. Wooley, On the Waring-Goldbach problem for eighth and higher powers, J.London Math. Soc. (to appear), preprint available as arXiv:1510.00982.[8] I. M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR15 (1937), 291–294.[9] T. D. Wooley, Vinogradov’s mean value theorem via efficient congruencing, Ann. of Math. (2) 175(2012), no. 3, 1575–1627.[10] T. D. Wooley, Multigrade efficient congruencing and Vinogradov’s mean value theorem, Proc. LondonMath. Soc. (3) 111 (2015), no. 3, 519–560.[11] L. Zhao, On the Waring–Goldbach problem for fourth and sixth powers, Proc. London Math. Soc. (3)108 (2014), no. 6, 1593–1622.Department of Mathematics, Towson University, Towson, MD 21252, USAE-mail address: akumchev@towson.eduSchool of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UKE-mail address: matdw@bristol.ac.uk6

English

On the Waring–Goldbach problem for seventh and higher powers

                          Kumchev, A., & Wooley, T. D. (2017). On the Waring–Goldbach problemfor seventh and higher powers. Monatshefte für Mathematik, 183(2),303–310. DOI: 10.1007/s00605-016-0936-7Peer reviewed versionLink to published version (if available):10.1007/s00605-016-0936-7Link to publication record in Explore Bristol ResearchPDF-documentThis is the accepted author manuscript (AAM). The final published version (version of record) is available onlinevia Springer Verlag at http://doi.org/10.1007/s00605-016-0936-7. Please refer to any applicable terms of use ofthe publisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlON THE WARING–GOLDBACH PROBLEMFOR SEVENTH AND HIGHER POWERSANGEL V. KUMCHEV AND TREVOR D. WOOLEYAbstract. We apply recent progress on Vinogradov’s mean value theorem to improvebounds for the function H(k) in the Waring–Goldbach problem. We obtain new results forall exponents k ≥ 7, and in particular establish that for large k one hasH(k) ≤ (4k − 2) log k − (2 log 2− 1)k − 3.1. IntroductionIn our recent work [7], we reported on the consequences for the Waring-Goldbach problemof recent progress on Vinogradov’s mean value theorem based on efficient congruencing(see, for example, [9, 10]). We now revisit our analysis in order to incorporate the latestdevelopments stemming from work of Bourgain, Demeter and Guth [1]. We first recall thedefinition of the function H(k) associated with the Waring-Goldbach problem. Consider anatural number k and prime number p, and define θ = θ(k, p) to be the integer with pθ|kbut pθ+1 - k, and γ = γ(k, p) byγ(k, p) ={θ + 2, when p = 2 and θ > 0,θ + 1, otherwise.We then put K(k) =∏(p−1)|k pγ, and denote by H(k) the least integer s such that everysufficiently large positive integer congruent to s modulo K(k) may be written as the sum ofs k-th powers of prime numbers.Improving on the bound H(k) ≤ k(4 log k + 2 log log k + O(1)), as k → ∞, due to Hua[3, 4], we recently showed that H(k) ≤ (4k − 2) log k + k − 7. The improved bound that wenow present in this note saves roughly (2 log 2)k further variables.Theorem 1. When k is large, one has H(k) ≤ (4k − 2) log k − (2 log 2− 1)k − 3.For small values of k one has the boundsH(1) ≤ 3, H(2) ≤ 5, H(3) ≤ 9, H(4) ≤ 13, H(5) ≤ 21, H(6) ≤ 32, H(7) ≤ 46,as a consequence of work of Vinogradov [8], Hua [2], Kawada and the second author [5], thefirst author [6], and Zhao [11]. For larger values of k, we recently established thatH(8) ≤ 61, H(9) ≤ 75, H(10) ≤ 89, H(11) ≤ 103, H(12) ≤ 117,H(13) ≤ 131, H(14) ≤ 147, H(15) ≤ 163, H(16) ≤ 178,H(17) ≤ 194, H(18) ≤ 211, H(19) ≤ 227, H(20) ≤ 244.We now obtain the following bounds for H(k) when 7 ≤ k ≤ 20.2000 Mathematics Subject Classification. 11P32, 11L20, 11P05, 11P55.1Theorem 2. Let 7 ≤ k ≤ 20. Then H(k) ≤ s(k), where s(k) is defined by Table 1.k 7 8 9 10 11 12 13 14 15 16 17 18 19 20s(k) 45 57 69 81 93 107 121 134 149 163 177 193 207 223Table 1. Upper bounds for H(k) when 7 ≤ k ≤ 20Our proof of Theorems 1 and 2 proceeds by directly incorporating the refinements availablevia [1] into our previous methods from [7]. We record improved estimates for Weyl sums in§2, both pointwise bounds and mean value estimates. Then, in §3, we indicate how to refineour previous bounds for H(k) using these bounds, thereby establishing Theorems 1 and 2.Throughout this paper, the letter ε denotes a sufficiently small positive number. Wheneverε occurs in a statement, we assert that the statement holds for each positive ε, and anyimplied constant in such a statement is allowed to depend on ε. The letter p, with orwithout subscripts, is reserved for prime numbers. We also write e(x) for exp(2piix), and(a, b) for the greatest common divisor of a and b. Finally, for real numbers θ, we denote bybθc the largest integer not exceeding θ, and by dθe the least integer no smaller than θ.2. Auxiliary estimates for exponential sumsWe refine the work of [7, §§2 and 3] by incorporating recent progress on Vinogradov’smean value theorem due to Bourgain, Demeter and Guth [1]. Recall the classical Weyl sumfk(α;X) =∑X<x≤2Xe(αxk),in which we suppose that k ≥ 2 is an integer and α is real. When k ≥ 3 is an integer, wedefine σk by means of the relation(2.1) σ−1k = min{2k−1, k(k − 1)}.Also, for k ≥ 3, we define the multiplicative function wk(q) by takingwk(puk+v) ={kp−u−1/2, when u ≥ 0 and v = 1,p−u−1, when u ≥ 0 and 2 ≤ v ≤ k.Lemma 2.1. Suppose that k ≥ 3. Then either one has fk(α;X) X1−σk+ε, or there existintegers a and q such that 1 ≤ q ≤ Xkσk , (a, q) = 1 and |qα− a| ≤ X−k+kσk , in which casefk(α;X) wk(q)X1 +Xk|α− a/q| +X1/2+ε.Proof. One may apply the argument of the proof of [7, Lemma 2.1], noting only that the re-finement of [10, Theorem 11.1] that follows by employing the bounds recorded in [1, Theorem1.1] permits the use of the exponent σk with the revised definition (2.1) presented above. We also require upper bounds for the corresponding Weyl sum over prime numbers,gk(α;X) =∑X<p≤2Xe(αpk),and these we summarise in the next lemma.2Lemma 2.2. Suppose that k ≥ 4 and X2σk/3 ≤ P ≤ X9/20. Then either one has the boundgk(α;X)  X1−σk/3+ε, or else there exist integers a and q such that 1 ≤ q ≤ P , (a, q) = 1and |qα− a| ≤ PX−k, in which case(2.2) gk(α;X) X1+ε(q +Xk|qα− a|)1/2 .Proof. One may follow the argument of the proof of [7, Lemma 2.2], noting that the refine-ment to the exponent σk made available via (2.1) as exhibited in Lemma 2.1. In order to describe our critical mean-value estimate, we introduce a set of admissibleexponents for kth powers as follows. Let t = tk and u = uk be positive integers to be fixedin due course. Put θ = 1− 1/k, and define(2.3) λi = (θ + σk−1/k)i−1 (1 ≤ i ≤ u+ 1).Then define λu+2, . . . , λu+t by puttingλu+2 =k2 − θt−3k2 + k − kθt−3λu+1,(2.4)λu+j =k2 − k − 1k2 + k − kθt−3 θj−3λu+1 (3 ≤ j ≤ t),(2.5)and then write(2.6) Λ = λ1 + . . .+ λt+u.Lemma 2.3. Let k, t and u be positive integers with k ≥ 3 and t ≥ ⌊12(k + 3)⌋, and let wbe a non-negative integer. Define the exponents λj and Λ by means of (2.3)-(2.6), and putη = max{0, k − Λ− 2wσk}. Then when N is sufficiently large, one has∫ 10|gk(α;N)|2wt+u∏j=1∣∣gk(α;Nλj)∣∣2 dα N2Λ+2w−k+η+ε.Proof. This is [7, Lemma 3.3], modified to reflect the improved Weyl exponent (2.1) asexhibited in Lemmata 2.1 and 2.2. 3. The upper bound for H(k)An upper bound for H(k) follows by combining the mean value estimate supplied byLemma 2.3 with the Weyl-type estimate stemming from Lemma 2.2.Lemma 3.1. Let k, t and u be positive integers with k ≥ 3 and t ≥ ⌊12(k + 3)⌋. Define theexponent Λ by means of (2.6), and put v = b(k−Λ)/(2σk)c and η∗ = k−Λ− 2vσk. Finally,defineh =1, when 0 ≤ η∗ < 12σk,2, when 12σk ≤ η∗ < σk,3, when σk ≤ η∗ < 2σk.Suppose in addition that 2(t + u + v) + h ≥ 3k + 1 and, when h ∈ {1, 2}, that either v ≥ 3or η∗ < hσk/3. ThenH(k) ≤ 2(t+ u+ v) + h.3Proof. This is [7, Lemma 4.1], modified to reflect the improved Weyl exponent (2.1) asexhibited in Lemma 2.1-2.3. We establish Theorems 1 and 2 by applying Lemma 3.1. Recall (2.3)-(2.6), and writeσ = σk−1 and φ = θ + σ/k. Then, just as in the discussion of [7, §5], one hask − Λ = − kσ1− σ +(k2(k + 1)σ + θt−3((k3 − 3k2 + k + 2)− σ(k3 − 2k2 + k + 2))(k2 + k − kθt−3)(1− σ))φu.The proof of Theorem 2. Let k be an integer with 7 ≤ k ≤ 20, and define t = tk, u = uk,v = vk and h = hk by means of Table 2. Then by application of a simple computer program,one confirms the validity of the hypotheses of Lemma 3.1. Thus H(k) ≤ 2(t + u + v) + h.Indeed, with h∗k defined as in Table 3, one finds that for each k one has 2η∗/σk < h∗k. Wenote in this context that the entries in this table have been rounded up in the final decimalplace presented. This completes our proof of Theorem 2.k 7 8 9 10 11 12 13 14 15 16 17 18 19 20tk 7 12 17 10 13 10 24 19 30 17 25 18 29 37uk 13 12 14 25 29 37 28 42 41 60 56 74 66 63vk 2 4 3 5 4 6 8 5 3 4 7 4 8 11hk 1 1 1 1 1 1 1 2 1 1 1 1 1 1Table 2. The values of tk, uk, vk and hk for 7 ≤ k ≤ 20k 7 8 9 10 11 12 13h∗k 0.44643 0.22927 0.02678 0.00739 0.97975 0.00042 0.08628k 14 15 16 17 18 19 20h∗k 1.94435 0.03925 0.01091 0.39085 0.00541 0.52855 0.00043Table 3. The values of h∗k for 7 ≤ k ≤ 20As we remarked in [7, §5], the non-monotonicity in the values of tk, uk and vk recorded inTable 2 is a consequence of the fact that θ and φ are close in size, and thus the optimisationis sensitive only to the sum tk + uk rather than the individual values of tk and uk.The proof of Theorem 1. We adapt the proof of [7, Theorem 1], supposing throughout thatk is sufficiently large. Put t = tk and u = uk, wheretk =⌈12k log k⌉and uk = dk(2 log k − log 2)e − t.It is convenient to define γ = dk(2 log k − log 2)e − k(2 log k − log 2). Also, we writeτ =1k(k − 1) and σ =1(k − 1)(k − 2) ,4so that σk = τ and σk−1 = σ. Our earlier formula for k − Λ now takes the shapek − Λ = − kσ1− σ +(k2(k + 1)(k − 1)3σ + θtk3(k3 − 3k2 +O(k))(k − 1)3(k2 + k − kθt−3)(1− σ))φu.As in the corresponding proof of [7, Theorem 1], one finds thatθt = e−t/k(1− log k4k+O(k−3/2)) k−1/2.Also, sincelog φ = log(1− 1− σk)= −1k− 12k2+O(1k3),one discerns thatφu = e−u/k(1− 3 log k − 2 log 24k+O(k−3/2)) k−3/2.Consequently,k − Λ = − kσ1− σ +(k − 1 +O(k−1/2)) θtφu +O(k−5/2),whereθtφu = e−(t+u)/k(1− 2 log k − log 22k+O(k−3/2))= e−γ/k(2k2− 2 log k − log 2k3+O(k−7/2)).Sinceστ=k(k − 1)(k − 1)(k − 2) = 1 +2k+O(1k2),we find thatk − Λ2τ= −12(k + 2) +e−γ/kk(k − 1)k3(k − 1)(k − log k + 12log 2) +O(k−1/2)= −12(k + 2) + (k − log k + 12log 2− 2) (1− γ/k) +O(k−1/2)= 12k − log k − 3 + 12log 2− γ +O(k−1/2).Put v = b(k − Λ)/(2τ)c, set η∗ = k − Λ − 2vτ , and define h as in the statement of Lemma3.1. Then one has 0 ≤ η∗ < 2τ , and in all circumstances one may confirm that2v + h =k − Λ− η∗τ+ h ≤ k − Λτ+ 2 ≤ k − 2 log k − 4 + log 2− 2γ +O(k−1/2).Since2(t+ u+ v) + h ≤ 2(2k log k − k log 2 + γ) + k − 2 log k − 4 + log 2− 2γ +O(k−1/2),we therefore conclude from Lemma 3.1 thatH(k) ≤ (4k − 2) log k − (2 log 2− 1)k − 4 + log 2 +O(k−1/2).We have assumed k to be sufficiently large, and thus we have established the boundH(k) ≤ (4k − 2) log k − (2 log 2− 1)k − 3.5This completes the proof of Theorem 1. References[1] J. Bourgain, C. Demeter and L. Guth, Proof of the main conjecture in Vinogradov’s mean value theoremfor degrees higher than three, preprint available as arXiv:1512.01565.[2] L. K. Hua, Some results in prime number theory, Quart. J. Math. Oxford Ser. 9 (1938), 68–80.[3] L. K. Hua, Additive Primzahltheorie, B. G. Teubner Verlagsgesellschaft, Leipzig, 1959.[4] L. K. Hua, Additive Theory of Prime Numbers, American Mathematical Society, Providence, RI, 1965.[5] K. Kawada and T. D. Wooley, On the Waring–Goldbach problem for fourth and fifth powers, Proc.London Math. Soc. (3) 83 (2001), no. 1, 1–50.[6] A. Kumchev, The Waring–Goldbach problem for seventh powers, Proc. Amer. Math. Soc. 133 (2005),no. 10, 2927-2937.[7] A. V. Kumchev and T. D. Wooley, On the Waring-Goldbach problem for eighth and higher powers, J.London Math. Soc. (to appear), preprint available as arXiv:1510.00982.[8] I. M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR15 (1937), 291–294.[9] T. D. Wooley, Vinogradov’s mean value theorem via efficient congruencing, Ann. of Math. (2) 175(2012), no. 3, 1575–1627.[10] T. D. Wooley, Multigrade efficient congruencing and Vinogradov’s mean value theorem, Proc. LondonMath. Soc. (3) 111 (2015), no. 3, 519–560.[11] L. Zhao, On the Waring–Goldbach problem for fourth and sixth powers, Proc. London Math. Soc. (3)108 (2014), no. 6, 1593–1622.Department of Mathematics, Towson University, Towson, MD 21252, USAE-mail address: akumchev@towson.eduSchool of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UKE-mail address: matdw@bristol.ac.uk6

On the Waring–Goldbach problem for seventh and higher powers

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