On upper bounds of chalk and hua for exponential sums

Proceedings of the American Mathematical Society

Volumn 129, Issue 9, 2001, Pages 2505-2516

  Cochrane, Todd ^a   Zheng, Zhiyong ^b  

^a KANSAS STATE UNIVERSITY   (United States)

^b TSINGHUA UNIVERSITY   (China)

Author keywords

Exponential sums

Indexed keywords

EID: 23044526693     PISSN: 00029939     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (48)

1. N.M. Akulinichev, Estimates for rational trigonometric sums of a special type, Doklady Acad. Sci. USSR 161 (1965), 743-745.
2. English transi, in Doklady 161, no. 4 (1965), 480-482.
3. J.H.H. Chalk, On Hua's estimate for exponential sums, Mathematika 34 (1987), 115-123. MR 89d:11067
4. J.R. Chcn, On the representation of natural numbers as a sum of terms of the form x(x + l)...(x + k- l)/k\, Acta Math. Sin. 8 (1958), 253-257.
5. -, On Professor Hua's estimate of exponential sums, Sci. Sinica 20 (1977), 711-719. MR 58:542
6. J.R. Chen and C. Pan, Analytic number theory in China I, in Number Theory and Its Applications in China, Contemp. Math. 77, Amer. Math. Soc. (1988), 1-17. MR 90c:11060
7. T. Cochrane and Z. Zheng, Pure and mixed exponential sums, Acta Arithmetica 91, no. 3 (1999), 249-278. MR 2000k:11093
8. _, Exponential sums with rational function entries, Acta Arithmetica 95, no. 1 (2000), 67-95. CMP 2001:02
9. P. Ding, An improvement to Chalk's estimation of exponential sums, Acta Arith. 59 no. 2 (1991), 149-155. MR 93a:11069
10. _, On a conjecture of Chalk, J. Number Theory 65 no. 2 (1997), 116-129. MR 98f:11086
11. G.H. Hardy and J.E. Littlewood, Some problems of "Partitio Numerorum"; I: A new solution of Waring's problem, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen Math.-phys. Klasse, (1920), 33-54.
12. D.R. Heath-Brown and S. Konyagin, New bounds for Gauss sums derived from kth powers, and for Heilbronn's exponential sum, Quart. J. Math. 51 (2000), 221-235. CMP 2000:14
13. L.K. Hua, On exponential sums, J. Chinese Math. Soc. 20 (1940), 301-312. MR 2:347h
14. _, On exponential sums, Sci. Record (Peking) (N.S.) 1 (1957), 1-4. MR 20:22
15. _, Additiv Primzahltheorie, Teubner, Leipzig (1959), 2-7.
16. M.N. Huxley, A note on polynomial congruences, Recent Progress in Analytic Number Theory, Vol. 1 (H. Halberstam and C. Hooley, eds.), Academic Press, London, 1981, 193-196. MR 83e:10005
17. A.A. Karatsuba, On bounds of complete trigonometrical sums, Matem. Zametki 1 no. 2 (1967), J99-208. English transi, in Math. Notes. Acad. Sci. USSR 1 (1967), 133-139.
18. S.V. Konyagin, Estimates for Gaussian sums and Waring's problem modulo a prime, Trudy Mat. Inst. Steklov 198 (1992), 111-124; translation in Proc. Steklov Inst. Math. 1994, 105-107. MR 96e:11122
19. _, Exponential sums over multiplicative groups of residues, preprint, (2000).
20. S.V. Konyagin and I.E. Shparlinski, On the distribution of residues of finitely generated multiplicative groups and their applications, Macquarie Mathematics Reports, Macquarie University, 1995.
21. _, Character sums with exponential functions and their applications, Cambridge Univ. Press, Cambridge, 1999. MR 2000h:11089
22. G. Lachaud, Bounds for exponential sums with invariant phase function, preprint.
23. W.K.A. Loh, Hua's Lemma, Bull. Austral. Math. Soc. 50, no. 3 (1994), 451-458. MR 95i:11091
24. J.H. Loxton and R.A. Smith, On Hua's estimate for exponential sums, J. London Math. Soc. (2), 26 (1982), 15-20. MR 84c:10033
25. J.H. Loxton and R.C. Vaughan, The estimation of complete exponential sums, Canad. Math. Bull. 28 no. 4 (1985), 442-454. MR 87c:11075
26. M. Lu, A note on the estimate of a complete rational trigonometric sum, Acta Math. Sin. 27 (1984), 817-823. MR 87a:11075
27. _, The estimate of complete trigonometric sums, Sci. Sin. 28, no. 6, (1985), 561-578. MR 87h:11078
28. _, A note on complete trigonometric sums for prime powers, Sichuan Daxue Xuebao 26 (1989), 156-159. MR 91g:11090
29. H.L. Montgomery, R.C. Vaughan and T.D. Wooley, Some remarks on Gauss suras associated with kth powers, Math. Proc. Cambridge Philos. Soc. 118, no. 1, (1995), 21-33. MR 96e:11110
30. L.J. Mordell, On a sum analogous to a Gauss's sum, Quart. J. Math., 3 (1932), 161-167.
31. G.L. Mullen and I.E. Shparlinski, Open problems and conjectures in finite fields, in Finite Fields and Applications (Glasgow, 1995), London Math. Soc. Lecture Note Series, No. 233, S. Cohen and H. Niederreiter, eds., Cambridge University Press, Cambridge, 1996, 243-268. MR 97m:11145
32. V.I. Necaev, An estimate of a complete rational trigonometric sum, Mat. Zametki 17 (1975), 839-849; English translation in Math. Notes 17 (1975). MR 53:5501
33. _, On the least upper bound on the modulus of complete trigonometric sums of degrees three and four, Investigations in number theory (Russian), Saratov. Gos. Univ., Saratov, (1988), 71-76. MR 91f:11059
34. V.I. Necaev and V.L. Topunov, Estimation of the modulus of complete rational trigonometric sums of degree three and four, Trudy Mat. Inst. Steklov, 158 (1981), 125-129; English translation in Proceedings of the Steklov Institute of Mathematics 1983, no. 4, Analytic number theory, mathematical analysis and their applications, Amer. Math. Soc., 135-140. MR 831:10048
35. M. Qi and P. Ding, Estimate of complete trigonometric sums, Kexue Tongbao 29 (1984), 1567-1569. MR 87b:11082a
36. _, On estimate of complete trigonometric sums, China Ann. Math. B6 (1985), 110-120. MR 87b:11082b
37. _, Further estimate of complete trigonometric sums, J. Tsinghua Univ. 29, no. 6, (1989), 74-85. MR 91m:11061
38. G. Sdndor, Über die Anzahl der Lösungen einer Kongruenz, Acta Math. 87 (1952), 13-17. MR 13:913d
39. W.M. Schmidt and C.L. Stewart, Congruences, trees andp-adic integers, Trans. Amer. Math. Soc. 349, no. 2, (1997), 605-639. MR 97e:11045
40. I.E. Shparlinski, On bounds of Gaussian sums, Matem. Zametki, 50 (1991), 122-130 (in Russian).
41. _, Computational and Algorithmic Problems in Finite Fields, Klmver Academic Pub., Boston, (1992). MR 94j:11122
42. _, On Gaussian sums for finite fields and elliptic curves, Proc. 1-st French-Soviet Workshop on Algebraic Coding, Paris, 1991, Lect. Notes in Computer Sci., 537 (1992), 5-15. MR 95c:11146
43. R.A. Smith, Estimates for exponential sums, Proc. Amer. Math. Soc. 79, no. 3, (1980), 365368. MR 81k:10059
44. S.B. Steckin, Estimate of a complete rational trigonometric sum, Proc. Steklov Inst. 143 (1977), 188-220, English translation, A.M.S. Issue 1 (1980), 201-220. MR 58:543
45. C.L. Stewart, On the number of solutions of polynomial congruences and Thue equations, 3. Amer. Math. Soc. 4, no. 4, (1991), 793-835. MR 92j:11032
46. R.C. Vaughan, The Hardy-Littlewood method, Cambridge Univ. Press, New York, (1981). MR 84b:10002
47. A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204-207. MR 10:234e
48. M. Zhang and Y. Hong, On the maximum modulus of complete trigonometric sums, Acta Math. Sinica, New Series 3, no. 4 (1987), 341-350. MR 89d:110G8