Upper Bounds on Character Sums with Rational Function Entries

Acta Mathematica Sinica, English Series

Volumn 19, Issue 2, 2003, Pages 327-338

  Cochrane, Todd ^a   Liu, Chun Lei ^b   Zheng, Zhi Yong ^c  

^a KANSAS STATE UNIVERSITY   (United States)

^b ZHENGZHOU UNIVERSITY   (China)

^c TSINGHUA UNIVERSITY   (China)

Author keywords

Character sums; Exponential sums

Indexed keywords

EID: 1442280647     PISSN: 14398516     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10114-002-0230-6     Document Type: Article

References (31)

1. Weil, A.: On some exponential sums. Proc. Nat. Acad. Sci., U.S.A., 34, 204-207 (1948)
2. Schmidt, W. M.: Equations over finite fields. L. N. M. 536, Springer-Verlag, Berlin (1976)
3. Stepanov, S. A.: Arithmetic of Algebraic Curves, English translation, Monographs in Contemporary Mathematics, Consultants Bureau, New York (1994)
4. Li, W. C. W.: Number Theory with Applications, World Scientific Pub., Singapore (1996)
5. Ismoilov, D.: Estimate of a character sum of polynomials. Dokl. Acad. Nauk Tadjhik SSR, (Russian) 29, 567-571 (1986)
6. Ismoilov, D.: Estimates for character sums of polynomials and rational functions, Construction Methods and Algorithms in Number Theory, Abstracts of Reports. Minsk, (Russian) 62 (1989)
7. Ismoilov, D.: Estimates of complete character sums of polynomials. Trudy Mat. Inst. Steklov, (Russian) 200, 171-186 (1991); English transl. in Proc. Steklov Inst. Math., 200, 189-203 (1993)
8. Ismoilov, D.: Estimates of complete character sums of polynomials. Trudy Mat. Inst. Steklov, (Russian) 200, 171-186 (1991); English transl. in Proc. Steklov Inst. Math., 200, 189-203 (1993)
9. Hua, L. K.: On exponential sums. J. Chinese Math. Soc., 20, 301-312 (1940)
10. Hua, L. K.: On exponential sums, Sci. Record, (Peking) (N. S.) 1, 1-4 (1957)
11. Hua, L. K.: Additive Primzahltheorie, Teubner, Leipzig, 2-7 (1959)
12. Cochrane, T., Zheng, Z.: Pure and mixed exponential sums. Acta Arithmetica, 91(3), 249-278 (1999)
13. Cochrane, T., Zheng, Z.: Exponential sums with rational function entries. Acta Arithmetica, XCV(1), 67-95 (2000)
14. Cochrane, T., Zheng, Z.: On upper bounds of Chalk and Hua for exponential sums. Proceedings of A. M. S., 129, 2505-2516 (2001)
15. Ismoilov, D.: Lower bounds on character sums of polynomials with respect to a composite modulus. Dokl. Acad. Nauk Tadzhik SSR, (Russian), 33, 501-505 (1990)
16. Ismoilov, D.: A lower bound estimate for complete sums of characters of polynomials and rational functions. Acta Math. Sinica, New Series, 9, 90-99 (1993)
17. Liu, C.: Character sums of rational functions. Sci. China, Ser. A, 38(2), 182-187 (1995)
18. Liu, C.: Dirichlet character sums. Acta Arith., 88(4), 299-309 (1999)
19. Cochrane, T.: Mixed exponential sums modulo prime powers. preprint
20. Ismoilov, D.: Estimate of a character sum of rational functions. Dokl. Acad. Nauk Tadjhik SSR, (Russian), 29, 635-639 (1986)
21. Ismoilov, D.: On lower bounds of character sums of rational functions with respect to a composite modulus. Vestnik Tadzhik. Gos. Univ. Mat., (Russian), 5, 27-32 (1990)
22. Ismoilov, D.: Estimates for complete trigonometric sums, Number Theory and Analysis. Trudy Mat. Inst. Steklov, (Russian), 207, 153-171 (1994); English transl. in Proc. Steklov Inst. Math., 207, 137-153 (1995)
23. Ismoilov, D.: Estimates for complete trigonometric sums, Number Theory and Analysis. Trudy Mat. Inst. Steklov, (Russian), 207, 153-171 (1994); English transl. in Proc. Steklov Inst. Math., 207, 137-153 (1995)
24. Liu, C.: Dirichlet character sums of rational functions. preprint (2000)
25. Chalk, J. H. H.: On Hua's estimate for exponential sums. Mathematika, 34, 115-123 (1987)
26. Ding, P.: An improvement to Chalk's estimation of exponential sums. Acta Arith., 59(2), 149-155 (1991)
27. Ding, P.: On a conjecture of Chalk. J. Number Theory, 65(2), 116-129 (1997)
28. Loh, W. K. A.: Hua's Lemma. Bull. Australian Math. Soc., 50(3), 451-458 (1994)
29. Postnikov, A. G.: On a character sum modulo a prime power of a prime. Izv. Acad. Nauk SSR, Ser. Mathem., (Russian), 19, 11-16 (1955)
30. Loxton, J. H., Vaughan, R. C.: The estimation of complete exponential sums. Canad. Math. Bull., 28(4), 442-454 (1985)
31. Salié, H.: Über die Kloostermanschen summen S(u,v;q). Math. Zeit., 34, 91-109 (1931)