ArticleOriginal scientific text

Title

A character-sum estimate and applications

Authors 1

Affiliations

  1. 94 Thornton Road, Bangor, Maine 04401-3336, U.S.A.

Bibliography

  1. N. C. Ankeny, The least quadratic nonresidue, Ann. of Math. 55 (1952), 65-72.
  2. E. Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), 355-380.
  3. E. Bach and L. Huelsbergen, Statistical evidence for small generating sets, Math. Comp. 61 (1993), 69-82.
  4. R. Bellman and B. Kotkin, On the numerical solution of a differential-difference equation arising in analytic number theory, Math. Comp. 16 (1962), 473-475.
  5. N. G. de Bruijn, The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.) 15 (1951), 25-32.
  6. A. A. Buchštab [A. A. Bukhshtab], On those numbers in an arithmetic progression all prime factors of which are small in order of magnitude, Dokl. Akad. Nauk SSSR (N.S.) 67 (1949), 5-8 (in Russian).
  7. D. A. Burgess, On character sums and L-series, Proc. London Math. Soc. (3) 12 (1962), 193-206.
  8. D. A. Burgess, On character sums and L-series. II, Proc. London Math. Soc. 13 (1963), 524-536.
  9. D. A. Burgess, A note on the distribution of residues and non-residues, J. London Math. Soc. 38 (1963), 253-256.
  10. D. A. Burgess, The character sum estimate with r = 3, J. London Math. Soc. (2) 33 (1986), 219-226.
  11. R. J. Burthe, Jr., Upper bounds for least witnesses and generating sets, Acta Arith. 80 (1997), 311-326.
  12. J.-M.-F. Chamayou, A probabilistic approach to a differential-difference equation arising in analytic number theory, Math. Comp. 27 (1973), 197-203.
  13. H. Davenport and P. Erdős, The distribution of quadratic and higher residues, Publ. Math. Debrecen 2 (1952), 252-265.
  14. P. D. T. A. Elliott, Some notes on kth power residues, Acta Arith. 14 (1968), 153-162.
  15. P. D. T. A. Elliott, Extrapolating the mean-values of multiplicative functions, Nederl. Akad. Wetensch. Proc. Ser. A 92 (1989), 409-420.
  16. P. D. T. A. Elliott, Some remarks about multiplicative functions of modulus ≤ 1, in: Analytic Number Theory (Allerton Park, Ill., 1989), Progr. Math. 85, Birkhäuser Boston, Boston, Mass., 1990, 159-164.
  17. E. Fouvry et G. Tenenbaum, Entiers sans grand facteur premier en progressions arithmétiques, Proc. London Math. Soc. (3) 63 (1991), 449-494.
  18. S. W. Graham and C. J. Ringrose, Lower bounds for least quadratic nonresidues, in: Analytic Number Theory (Allerton Park, Ill., 1989), Progr. Math. 85, Birkhäuser Boston, Boston, Mass., 1990, 269-309.
  19. H. Hasse, Vorlesungen über Zahlentheorie, 2nd ed., Springer, Berlin, 1964.
  20. D. G. Hazlewood, Sums over positive integers with few prime factors, J. Number Theory 7 (1975), 189-207.
  21. A. Hildebrand and G. Tenenbaum, Integers without large prime factors, J. Théor. Nombres Bordeaux 5 (1993), 411-484.
  22. J. H. Jordan, The distribution of cubic and quintic non-residues, Pacific J. Math. 16 (1966), 77-85.
  23. J. H. Jordan, The distribution of kth power residues and non-residues, Proc. Amer. Math. Soc. 19 (1968), 678-680.
  24. J. H. Jordan, The distribution of kth power non-residues, Duke Math. J. 37 (1970), 333-340.
  25. G. Kolesnik and E. G. Straus, On the first occurrence of values of a character, Trans. Amer. Math. Soc. 246 (1978), 385-394.
  26. B. V. Levin and A. S. Faĭnleĭb, Application of some integral equations to problems of number theory, Uspekhi Mat. Nauk 22 (1967), no. 3, 119-197 (in Russian); English transl.: Russian Math. Surveys 22 (1967), no. 3, 119-204.
  27. J. van de Lune and E. Wattel, On the numerical solution of a differential-difference equation arising in analytic number theory, Math. Comp. 23 (1969), 417-421.
  28. H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer, Berlin, 1971.
  29. K. K. Norton, Upper bounds for kth power coset representatives modulo n, Acta Arith. 15 (1969), 161-179.
  30. K. K. Norton, On the distribution of kth power residues and non-residues modulo n, J. Number Theory 1 (1969), 398-418.
  31. K. K. Norton, Numbers with small prime factors, and the least kth power non-residue, Mem. Amer. Math. Soc. 106 (1971).
  32. K. K. Norton, On the distribution of power residues and non-residues, J. Reine Angew. Math. 254 (1972), 188-203.
  33. K. K. Norton, On character sums and power residues, Trans. Amer. Math. Soc. 167 (1972), 203-226.
  34. K. K. Norton, Bounds for sequences of consecutive power residues. I, in: Analytic Number Theory, Proc. Sympos. Pure Math. 24, Amer. Math. Soc., Providence, R.I., 1973, 213-220.
  35. F. Pappalardi, On minimal sets of generators for primitive roots, Canad. Math. Bull. 38 (1995), 465-468.
  36. G. Tenenbaum, Cribler les entiers sans grand facteur premier, Philos. Trans. Roy. Soc. London Ser. A 345 (1993), 377-384.
  37. T. Z. Xuan, Integers with no large prime factors, Acta Arith. 69 (1995), 303-327.
Pages:
51-78
Main language of publication
English
Received
1997-06-10
Published
1998
Exact and natural sciences