A character-sum estimate and applications

Karl K. Norton

Acta Arithmetica (1998)

  • Volume: 85, Issue: 1, page 51-78
  • ISSN: 0065-1036

How to cite

top

Karl K. Norton. "A character-sum estimate and applications." Acta Arithmetica 85.1 (1998): 51-78. <http://eudml.org/doc/207154>.

@article{KarlK1998,
author = {Karl K. Norton},
journal = {Acta Arithmetica},
keywords = {character sum estimates; Burgess' inequality},
language = {eng},
number = {1},
pages = {51-78},
title = {A character-sum estimate and applications},
url = {http://eudml.org/doc/207154},
volume = {85},
year = {1998},
}

TY - JOUR
AU - Karl K. Norton
TI - A character-sum estimate and applications
JO - Acta Arithmetica
PY - 1998
VL - 85
IS - 1
SP - 51
EP - 78
LA - eng
KW - character sum estimates; Burgess' inequality
UR - http://eudml.org/doc/207154
ER -

References

top
  1. [1] N. C. Ankeny, The least quadratic nonresidue, Ann. of Math. 55 (1952), 65-72. Zbl0046.04006
  2. [2] E. Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), 355-380. Zbl0701.11075
  3. [3] E. Bach and L. Huelsbergen, Statistical evidence for small generating sets, Math. Comp. 61 (1993), 69-82. Zbl0784.11059
  4. [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. Zbl0106.10602
  5. [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. Zbl0043.06502
  6. [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. [7] D. A. Burgess, On character sums and L-series, Proc. London Math. Soc. (3) 12 (1962), 193-206. Zbl0106.04004
  8. [8] D. A. Burgess, On character sums and L-series. II, Proc. London Math. Soc. 13 (1963), 524-536. Zbl0123.04404
  9. [9] D. A. Burgess, A note on the distribution of residues and non-residues, J. London Math. Soc. 38 (1963), 253-256. Zbl0118.04703
  10. [10] D. A. Burgess, The character sum estimate with r = 3, J. London Math. Soc. (2) 33 (1986), 219-226. Zbl0593.10033
  11. [11] R. J. Burthe, Jr., Upper bounds for least witnesses and generating sets, Acta Arith. 80 (1997), 311-326. Zbl0880.11008
  12. [12] J.-M.-F. Chamayou, A probabilistic approach to a differential-difference equation arising in analytic number theory, Math. Comp. 27 (1973), 197-203. Zbl0252.65066
  13. [13] H. Davenport and P. Erdős, The distribution of quadratic and higher residues, Publ. Math. Debrecen 2 (1952), 252-265. Zbl0050.04302
  14. [14] P. D. T. A. Elliott, Some notes on kth power residues, Acta Arith. 14 (1968), 153-162. Zbl0199.36503
  15. [15] P. D. T. A. Elliott, Extrapolating the mean-values of multiplicative functions, Nederl. Akad. Wetensch. Proc. Ser. A 92 (1989), 409-420. Zbl0696.10041
  16. [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. [17] E. Fouvry et G. Tenenbaum, Entiers sans grand facteur premier en progressions arithmétiques, Proc. London Math. Soc. (3) 63 (1991), 449-494. Zbl0745.11042
  18. [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. [19] H. Hasse, Vorlesungen über Zahlentheorie, 2nd ed., Springer, Berlin, 1964. 
  20. [20] D. G. Hazlewood, Sums over positive integers with few prime factors, J. Number Theory 7 (1975), 189-207. Zbl0302.10037
  21. [21] A. Hildebrand and G. Tenenbaum, Integers without large prime factors, J. Théor. Nombres Bordeaux 5 (1993), 411-484. Zbl0797.11070
  22. [22] J. H. Jordan, The distribution of cubic and quintic non-residues, Pacific J. Math. 16 (1966), 77-85. Zbl0151.02701
  23. [23] J. H. Jordan, The distribution of kth power residues and non-residues, Proc. Amer. Math. Soc. 19 (1968), 678-680. Zbl0164.35102
  24. [24] J. H. Jordan, The distribution of kth power non-residues, Duke Math. J. 37 (1970), 333-340. 
  25. [25] G. Kolesnik and E. G. Straus, On the first occurrence of values of a character, Trans. Amer. Math. Soc. 246 (1978), 385-394. Zbl0399.10037
  26. [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. Zbl0204.06502
  27. [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. Zbl0176.46602
  28. [28] H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer, Berlin, 1971. Zbl0216.03501
  29. [29] K. K. Norton, Upper bounds for kth power coset representatives modulo n, Acta Arith. 15 (1969), 161-179. Zbl0177.06801
  30. [30] K. K. Norton, On the distribution of kth power residues and non-residues modulo n, J. Number Theory 1 (1969), 398-418. Zbl0185.10503
  31. [31] K. K. Norton, Numbers with small prime factors, and the least kth power non-residue, Mem. Amer. Math. Soc. 106 (1971). 
  32. [32] K. K. Norton, On the distribution of power residues and non-residues, J. Reine Angew. Math. 254 (1972), 188-203. Zbl0234.10033
  33. [33] K. K. Norton, On character sums and power residues, Trans. Amer. Math. Soc. 167 (1972), 203-226. Zbl0238.10023
  34. [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. [35] F. Pappalardi, On minimal sets of generators for primitive roots, Canad. Math. Bull. 38 (1995), 465-468. Zbl0840.11039
  36. [36] G. Tenenbaum, Cribler les entiers sans grand facteur premier, Philos. Trans. Roy. Soc. London Ser. A 345 (1993), 377-384. 
  37. [37] T. Z. Xuan, Integers with no large prime factors, Acta Arith. 69 (1995), 303-327. Zbl0819.11035

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.