The determination of Gauss sums
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- by Bruce C. Berndt and Ronald J. Evans PDF
- Bull. Amer. Math. Soc. 5 (1981), 107-129
References
- Chungming An, On values of exponential sums, Proc. Amer. Math. Soc. 52 (1975), 131–135. MR 376561, DOI 10.1090/S0002-9939-1975-0376561-3
- George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
- Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
- L. Auslander and R. Tolimieri, Is computing with the finite Fourier transform pure or applied mathematics?, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 847–897. MR 546312, DOI 10.1090/S0273-0979-1979-14686-X
- Raymond Ayoub, An introduction to the analytic theory of numbers, Mathematical Surveys, No. 10, American Mathematical Society, Providence, R.I., 1963. MR 0160743 6. P. Bachmann, Die Lehre von der Kreistheilung, Teubner, Leipzig, 1872.
- R. P. Bambah and S. Chowla, On the sign of the Gaussian sum, Proc. Nat. Inst. Sci. India 13 (1947), 175–176. MR 25491
- Klaus Barner, Zur Reziprozität quadratischer Charaktersummen in algebraischen Zahlkörpern, Monatsh. Math. 71 (1967), 369–384 (German). MR 228460, DOI 10.1007/BF01300643
- L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Information and Control 20 (1972), 158–175. MR 497284, DOI 10.1016/S0019-9958(72)90354-3
- Richard Bellman, A brief introduction to theta functions, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1961. MR 0125252, DOI 10.1017/s0025557200044491
- Bruce C. Berndt, On Gaussian sums and other exponential sums with periodic coefficients, Duke Math. J. 40 (1973), 145–156. MR 313202
- Bruce C. Berndt and S. Chowla, The reckoning of certain quartic and octic Gauss sums, Glasgow Math. J. 18 (1977), no. 2, 153–155. MR 457369, DOI 10.1017/S0017089500003190
- Bruce C. Berndt and Ronald J. Evans, Sums of Gauss, Jacobi, and Jacobsthal, J. Number Theory 11 (1979), no. 3, S. Chowla Anniversary Issue, 349–398. MR 544263, DOI 10.1016/0022-314X(79)90008-8
- Bruce C. Berndt and Ronald J. Evans, Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer, Illinois J. Math. 23 (1979), no. 3, 374–437. MR 537798
- Gudrun Beyer, Über eine Klasseneinteilung aller kubischen Restcharaktere, Abh. Math. Sem. Univ. Hamburg 19 (1954), no. 1-2, 115–116 (German). MR 63394, DOI 10.1007/BF02941559
- A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
- Maurizio Boyarsky, $p$-adic gamma functions and Dwork cohomology, Trans. Amer. Math. Soc. 257 (1980), no. 2, 359–369. MR 552263, DOI 10.1090/S0002-9947-1980-0552263-1
- D. M. Bressoud, Applications of Andrews’ basic Lauricella transformation, Proc. Amer. Math. Soc. 72 (1978), no. 1, 89–94. MR 486677, DOI 10.1090/S0002-9939-1978-0486677-1
- D. M. Bressoud, On the value of Gaussian sums, J. Number Theory 13 (1981), no. 1, 88–94. MR 602449, DOI 10.1016/0022-314X(81)90030-5 20. W. Burnside, On cyclotomic quinquisection, Proc. London Math. Soc. 14 (1915), 251-259. 21. F. S. Carey, Notes on the division of the circle, Quart. J. Math. 26 (1893), 322-371.
- L. Carlitz, A note on Gauss’ sum, Proc. Amer. Math. Soc. 7 (1956), 910–911. MR 80698, DOI 10.1090/S0002-9939-1956-0080698-4
- L. Carlitz, A note on Gauss’s sum, Matematiche (Catania) 23 (1968), 147–150. MR 240044
- J. W. S. Cassels, On the determination of generalized Gauss sums, Arch. Math. (Brno) 5 (1969), 79–84. MR 294266
- J. W. S. Cassels, On Kummer sums, Proc. London Math. Soc. (3) 21 (1970), 19–27. MR 266895, DOI 10.1112/plms/s3-21.1.19 26. J. W. S. Cassels, Review of E. D. Tabakova, "A numerical investigation of Kummer cubic sums", (Inst Appl. Math. USSR Acad. Sci., Moscow, preprint No. 98 (1973), 22 pages), Math. Comp. 29 (1975), 665-666.
- J. W. S. Cassels, Trigonometric sums and elliptic functions, Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976) Japan Soc. Promotion Sci., Tokyo, 1977, pp. 1–7. MR 0450209 28. A. Cauchy, Méthode simple et nouvelle pour la détermination complète des sommes alternées formées avec les racines primitives des équationes binomes, C. R. Acad. Sci. Paris 10 (1840), 560-572. 29. A. Cauchy, Méthode simple et nouvelle pour la détermination complète des sommes alternées formées avec les racines primitives des équationes binomes, J. de Math. 5 (1840), 154-168. (Same article as in reference 28.) 30. A. Cauchy, Oeuvres, I Série, t.v, Gauthier Villars, Paris, 1885. 31. A. Cayley, On the binomial equation x 1 = 0; trisection and quartisection, Proc. London Math. Soc. 11 (1879), 4-17. 32. A. Cauchy, The binomial equation x 1 = 0; quinquisection, Proc. London Math. Soc. 12 (1880), 15-16. 33. A. Cauchy, The binomial equation x 1 = 0; quinquisection, second note, Proc. London Math. Soc. 16 (1885), 61-63.
- K. Chandrasekharan, Introduction to analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 148, Springer-Verlag New York, Inc., New York, 1968. MR 0249348, DOI 10.1007/978-3-642-46124-8
- S. Chowla, The Riemann hypothesis and Hilbert’s tenth problem, Mathematics and its Applications, Vol. 4, Gordon and Breach Science Publishers, New York-London-Paris, 1965. MR 0177943 36. J. G. van der Corput, Zahlentheoretische Abschätzungen, Math. Ann. 84 (1921), 53-79.
- Harold Davenport, Multiplicative number theory, Lectures in Advanced Mathematics, No. 1, Markham Publishing Co., Chicago, Ill., 1967. Lectures given at the University of Michigan, Winter Term, 1966. MR 0217022 38. H. Davenport and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math. 172 (1934), 151-182. 39. R. Dedekind, Review of P. Bachmann, "Die Lehre von der Kreistheilung", Z. Math. Phys. 18 (1873), 14-24 (in Literaturzeitung).
- P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
- L. E. Dickson, Cyclotomy, Higher Congruences, and Waring’s Problem, Amer. J. Math. 57 (1935), no. 2, 391–424. MR 1507083, DOI 10.2307/2371217
- L. E. Dickson, H. H. Mitchell, H. S. Vandiver, and G. E. Wahlin, Algebraic numbers, Chelsea Publishing Co., New York, 1967. MR 0241349 43. P. G. L. Dirichlet, Ueber eine neue Anwendung bestimmter Integrale auf die Summation endlicher oder unendlicher Reihen, Abh. K. Preussischen Akad. Wiss., 1835, 391-407. 44. P. G. L. Dirichlet, Sur l’usage des intégrales définies dans la sommation des séries finies ou infinies, J. Reine Angew. Math. 17 (1837), 57-67. 45. P. G. L. Dirichlet, Recherches sur diverses applications de l’analyse infinitésimale à la théorie des nombres, J. Reine Angew. Math. 21 (1840), 134-155. 46. P. G. L. Dirichlet, Werke, Erster Band, Georg Reimer, Berlin, 1889. 47. P. G. L. Dirichlet, Vorlesungen über Zahlentheorie, 4th. ed., Friedrich Vieweg und Sohn, Braunschweig, 1894.
- Martin Eichler, Introduction to the theory of algebraic numbers and functions, Pure and Applied Mathematics, Vol. 23, Academic Press, New York-London, 1966. Translated from the German by George Striker. MR 0209258 49. G. Eisenstein, Beweis der allgemeinsten Reziprozitätsgesetze zwischen reellen und complexen Zahlen, Monatsber. Preuss. Akad. Wiss. Berlin, 1850, 189-198. 50. G. Eisenstein, Mathematische Werke, vol. II, Chelsea, New York, 1975.
- T. Estermann, On the sign of the Gaussian sum, J. London Math. Soc. 20 (1945), 66–67. MR 15412, DOI 10.1112/jlms/s1-20.2.66
- Ronald J. Evans, Generalizations of a theorem of Chowla on Gaussian sums, Houston J. Math. 3 (1977), no. 3, 343–349. MR 498491
- Ronald J. Evans, Unambiguous evaluations of bidecic Jacobi and Jacobsthal sums, J. Austral. Math. Soc. Ser. A 28 (1979), no. 2, 235–240. MR 550967, DOI 10.1017/S1446788700015731
- Ronald J. Evans, The $2^{r}$th power character of $2$, J. Reine Angew. Math. 315 (1980), 174–189. MR 564532, DOI 10.1515/crll.1980.315.174
- Ronald J. Evans, Bioctic Gauss sums and sixteenth power residue difference sets, Acta Arith. 38 (1980/81), no. 1, 37–46. MR 574123, DOI 10.4064/aa-38-1-37-46
- Ronald J. Evans, Twenty-fourth power residue difference sets, Math. Comp. 40 (1983), no. 162, 677–683. MR 689481, DOI 10.1090/S0025-5718-1983-0689481-4
- Ronald J. Evans, Rational reciprocity laws, Acta Arith. 39 (1981), no. 3, 281–294. MR 640916, DOI 10.4064/aa-39-3-281-294
- Ronald J. Evans, Pure Gauss sums over finite fields, Mathematika 28 (1981), no. 2, 239–248 (1982). MR 645104, DOI 10.1112/S0025579300010299
- Ronald J. Evans, Identities for products of Gauss sums over finite fields, Enseign. Math. (2) 27 (1981), no. 3-4, 197–209 (1982). MR 659148 60. R. Fricke, Lehrbuch der Algebra, Band I, Friedrich Vieweg und Sohn, Braunschweig, 1924.
- Carl-Erik Fröberg, New results on the Kummer conjecture, Nordisk Tidskr. Informationsbehandling (BIT) 14 (1974), 117–119. MR 337728, DOI 10.1007/bf01933125
- A. Fröhlich, Non-Abelian Jacobi sums, Number theory and algebra, Academic Press, New York, 1977, pp. 71–75. MR 0480432
- Carl Friedrich Gauss, Mathematisches Tagebuch, 1796–1814, Ostwalds Klassiker der Exakten Wissenschaften, Band 256, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1976 (German). Mit einer historischen Einführung von Kurt-R. Biermann; Durchgesehen und mit einem Vorwort und Anmerkungen versehen von Hans Wussing; Übersetzt aus dem Lateinischen von Elisabeth Schuhmann. MR 0490786 64. C. F. Gauss, Summatio quarumdam serierum singularium, Comm. soc. reg. sci. Gottingensis rec. 1 (1811). 65. C. F. Gauss, Theoria residuorum biquadraticorum, Comment. I, Comm. soc. reg. sci. Gottingensis rec. 6 (1828). 66. C. F. Gauss, Werke, K. Gesell. Wiss., Göttingen, 1876.
- Carl Friedrich Gauss, Disquisitiones arithmeticae, Yale University Press, New Haven, Conn.-London, 1966. Translated into English by Arthur A. Clarke, S. J. MR 0197380 68. A. Genocchi, Sulla formula sommatoria di Eulero, e sulla teorica de’residui quadratici, Ann. Sci. Mat. Fis. (Roma) 3 (1852), 406-436. 69. A. Genocchi, Note sur la théorie des résidus quadratiques, Mem. Cour. Savants Etrangers, Acad. Roy. Sci. Lettres Beaux Arts Belgique 25 (1851/53), 54 pp.
- J. C. Glashan, Quinquisection of the Cylotomic Equation, Amer. J. Math. 21 (1899), no. 3, 270–275. MR 1505800, DOI 10.2307/2369604 71. G. Gras, Sommes de Gauss sur les corps finis, Publ. Math. Besancon, 1977-78, 71 pp.
- Benedict H. Gross and Neal Koblitz, Gauss sums and the $p$-adic $\Gamma$-function, Ann. of Math. (2) 109 (1979), no. 3, 569–581. MR 534763, DOI 10.2307/1971226
- Helmut Hasse, Zahlentheorie, Akademie-Verlag, Berlin, 1969 (German). Dritte berichtigte Auflage. MR 0253972
- D. R. Heath-Brown and S. J. Patterson, The distribution of Kummer sums at prime arguments, J. Reine Angew. Math. 310 (1979), 111–130. MR 546667 75. E. Hecke, Reziprozitätsgesetz und Gausssche Summen in quadratischen Zahlkörpern, Nachr. Gesell. Wiss. Göttingen, Math.-Phys. Kl. 1919, 265-278.
- Erich Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1970 (German). Mit einer Vorbemerkung von B. Schoenberg, einer Anmerkung von Carl Ludwig Siegel, und einer Todesanzeige von Jakob Nielsen; Zweite durchgesehene Auflage. MR 0371577 77. E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, Chelsea, New York, 1948.
- Anna Helversen-Pasotto, L’identité de Barnes pour les corps finis, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 6, A297–A300 (French, with English summary). MR 476707
- Peter Henrici, Computational complex analysis, The influence of computing on mathematical research and education (Proc. Sympos. Appl. Math., Vol. 20, Univ. Montana, Missoula, Mont., 1973), Amer. Math. Soc., Providence, R.I., 1974, pp. 79–86. MR 0349957
- Kenneth Ireland and Michael I. Rosen, Elements of number theory. Including an introduction to equations over finite fields, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1972. MR 0554185
- Sonoko Ishimura, On Gaussian sums associated with a character of order 5 and a rational prime number $p\equiv 1 (\textrm {mod}\ 5)$, J. Tsuda College 8 (1976), 27–35. MR 463125
- Jean-René Joly, Équations et variétés algébriques sur un corps fini, Enseign. Math. (2) 19 (1973), 1–117 (French). MR 327723 83. A. Krazer, Lehrbuch der Thetafunktionen, Teubner, Leipzig, 1903. 84. L. Kronecker, Sur une formule de Gauss, J. de Math. 21 (1856), 392-395. 85. L. Kronecker, Über den vierten Gauss’schen Beweis des Reziprozitätsgesetzes für die quadratischen Reste, Monatsber, K. Preuss. Akad. Wiss. Berlin, (1880), 686-698, 854-860. 86. L. Kronecker, Summirung der Gaussschen Reihen $\sum _{h=0}^{h=n 1}\ e$ 87. L. Kronecker, Über die Dirichletsche Methode der Wertbestimmung der Gaussschen Reihen, Mitth. Math. Gesell. Hamburg 2 (1890), 32-36.
- Daniel S. Kubert and Serge Lang, Independence of modular units on Tate curves, Math. Ann. 240 (1979), no. 3, 191–210. MR 526842, DOI 10.1007/BF01362309
- Tomio Kubota, An application of the power residue theory to some abelian functions, Nagoya Math. J. 27 (1966), 51–54. MR 204412, DOI 10.1017/S0027763000011843
- Tomio Kubota, On a special kind of Dirichlet series, J. Math. Soc. Japan 20 (1968), 193–207. MR 228455, DOI 10.2969/jmsj/02010193
- Tomio Kubota, Some results concerning reciprocity and functional analysis, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 395–399. MR 0429748
- Tomio Kubota, Some results concerning reciprocity law and real analytic automorphic functions, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 382–395. MR 0340221 93. E. E. Kummer, Eine Aufgabe, betreffend die Theorie der kubischen Reste, J. Reine Angew. Math. 23 (1842), 285-286. 94. E. E. Kummer, De residuis cubicis disquisitiones nonnullae analyticae, J. Reine Angew. Math. 32 (1846), 341-359. 95. E. E. Kummer, Theorie der idealen Primfaktoren der complexen Zahlen, welche aus den Wurzeln der Gleichung ωn = 1 gebildet sind, wenn n eine zusammengesetzte Zahl ist, Math. Abh. K. Akad. Wiss. Berlin (1856), 1-47.
- Ernst Eduard Kummer, Collected papers, Springer-Verlag, Berlin-New York, 1975. Volume I: Contributions to number theory; Edited and with an introduction by André Weil. MR 0465760 97. E. Landau, Über das Vorzeichen der Gaussschen Summe, Nachr. Gesell. Wiss. Göttingen, Math.-Phys. KI. 1928, 19-20.
- Edmund Landau, Elementary number theory, Chelsea Publishing Co., New York, N.Y., 1958. Translated by J. E. Goodman. MR 0092794
- Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
- Serge Lang, Cyclotomic fields, Graduate Texts in Mathematics, Vol. 59, Springer-Verlag, New York-Heidelberg, 1978. MR 0485768, DOI 10.1007/978-1-4612-9945-5
- Serge Lang, Cyclotomic fields, Graduate Texts in Mathematics, Vol. 59, Springer-Verlag, New York-Heidelberg, 1978. MR 0485768, DOI 10.1007/978-1-4612-9945-5 102. V.-A. Lebesgue, Recherches sur les nombres, J. de Math. 3 (1838), 113-144. 103. V.-A. Lebesgue, Sommation de quelques séries, J. de Math. 5 (1840), 42-71. 104. V.-A. Lebesgue, Note sur les congruences, C. R. Acad. Sci. (Paris) 51 (1860), 9-13. 105. A.-M. Legendre, Théorie des nombres, t. II, 4th. ed., Firmin Didot Frères, Paris, 1830.
- Emma Lehmer, The quintic character of $2$ and $3$, Duke Math. J. 18 (1951), 11–18. MR 40338
- Emma Lehmer, On the location of Gauss sums, Math. Tables Aids Comput. 10 (1956), 194–202. MR 86092, DOI 10.1090/S0025-5718-1956-0086092-2
- Emma Lehmer and Leonard Carlitz, Advanced Problems and Solutions: Solutions: 4636, Amer. Math. Monthly 63 (1956), no. 8, 584–587. MR 1529423, DOI 10.2307/2310217 109. G. Libri, Mémoire sur la théorie des nombres, J. Reine Angew. Math. 9 (1832), 169-188. 110. G. Libri, Résponse de M. Libri aux observations de M. Liouville, C. R. Acad. Sci. (Paris) 10 (1840), 345-347. 111. E. Lindelöf, Le calcul des résidus, Chelsea, New York, 1947. 112. J. Liouville, Sur les deux derniers cahiers du Journal de M. Crelle, J. de Math. 3 (1838), 3-5. 113. J. Liouville, Observations sur une Note de M. Libri, C. R. Acad. Sci. (Paris) 10 (1840), 343-345.
- J. H. Loxton, Products related to Gauss sums, J. Reine Angew. Math. 268(269) (1974), 53–67. MR 344205, DOI 10.1515/crll.1974.268-269.53
- John H. Loxton, On the determination of Gauss sums, Séminaire Delange-Pisot-Poitou, 18e année: 1976/77, Théorie des nombres, Fasc. 2, Secrétariat Math., Paris, 1977, pp. Exp. No. 27, 12. MR 551351
- J. H. Loxton, Some conjectures concerning Gauss sums, J. Reine Angew. Math. 297 (1978), 153–158. MR 460250, DOI 10.1515/crll.1978.297.153
- J. Martinet, Character theory and Artin $L$-functions, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 1–87. MR 0447187 118. G. B. Mathews, Theory of numbers, second ed., Chelsea, New York. 119. C. R. Matthews, Gauss sums and elliptic functions, Ph. D. thesis, Cambridge, 1978.
- C. R. Matthews, Gauss sums and elliptic functions. I. The Kummer sum, Invent. Math. 52 (1979), no. 2, 163–185. MR 536079, DOI 10.1007/BF01403063
- C. R. Matthews, Gauss sums and elliptic functions. II. The quartic sum, Invent. Math. 54 (1979), no. 1, 23–52. MR 549544, DOI 10.1007/BF01391175
- R. J. McEliece, Irreducible cyclic codes and Gauss sums, Combinatorics (Proc. NATO Advanced Study Inst., Breukelen, 1974) Math. Centre Tracts, No. 55, Math. Centrum, Amsterdam, 1974, pp. 179–196. MR 0363657
- Andrew D. McGettrick, A result in the theory of Weierstrass elliptic functions, Proc. London Math. Soc. (3) 25 (1972), 41–54. MR 318523, DOI 10.1112/plms/s3-25.1.41
- Andrew D. McGettrick, On the biquadratic Gauss sum, Proc. Cambridge Philos. Soc. 71 (1972), 79–83. MR 289427, DOI 10.1017/s0305004100050301 125. F. Mertens, Ueber die Gaussischen Summen, Sitz. Berliner Akad., 1896, 217-219.
- Howard H. Mitchell, On the generalized Jacobi-Kummer cyclotomic function, Trans. Amer. Math. Soc. 17 (1916), no. 2, 165–177. MR 1501034, DOI 10.1090/S0002-9947-1916-1501034-9 127. L. J. Mordell, On a simple summation of the series $\sum _{s=0}^{n-1}e^{2s^2\pi i/n}$, Messenger of Math. 48 (1918), 54-56. 128. L. J. Mordell, The definite integral \[ \int łimits_ {-\infty }^ \infty {\tfrac {{e^ {ax^ 2 + bx} }}{{e^ ax + d}}da}\] and the analytic theory of numbers, Acta Math. 61 (1933), 323-360.
- L. J. Mordell, The sign of the Gaussian sum, Illinois J. Math. 6 (1962), 177–180. MR 138581, DOI 10.1215/ijm/1255631818
- Carlos Julio Moreno, Sur le problème de Kummer, Enseign. Math. (2) 20 (1974), 45–51 (French). MR 374057
- Joseph B. Muskat and Albert L. Whiteman, The cyclotomic numbers of order twenty, Acta Arith. 17 (1970), 185–216. MR 268151, DOI 10.4064/aa-17-2-185-216
- Gerald Myerson, Period polynomials and Gauss sums for finite fields, Acta Arith. 39 (1981), no. 3, 251–264. MR 640913, DOI 10.4064/aa-39-3-251-264
- Trygve Nagell, Introduction to number theory, 2nd ed., Chelsea Publishing Co., New York, 1964. MR 0174513
- Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, Monografie Matematyczne, Tom 57, PWN—Polish Scientific Publishers, Warsaw, 1974. MR 0347767
- J. von Neumann and H. H. Goldstine, A numerical study of a conjecture of Kummer, Math. Tables Aids Comput. 7 (1953), 133–134. MR 55784, DOI 10.1090/S0025-5718-1953-0055784-0
- H. Niederreiter, On the cycle structure of linear recurring sequences, Math. Scand. 38 (1976), no. 1, 53–77. MR 406918, DOI 10.7146/math.scand.a-11616
- Harald Niederreiter, Weights of cyclic codes, Information and Control 34 (1977), no. 2, 130–140. MR 449872, DOI 10.1016/S0019-9958(77)80009-0
- Harald Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), no. 6, 957–1041. MR 508447, DOI 10.1090/S0002-9904-1978-14532-7
- R. Odoni, On Gauss sums $(\textrm {mod}\ p^{n})$, $n\geq 2$, Bull. London Math. Soc. 5 (1973), 325–327. MR 327678, DOI 10.1112/blms/5.3.325
- S. J. Patterson, A cubic analogue of the theta series, J. Reine Angew. Math. 296 (1977), 125–161. MR 563068, DOI 10.1515/crll.1977.296.125
- S. J. Patterson, A cubic analogue of the theta series, J. Reine Angew. Math. 296 (1977), 125–161. MR 563068, DOI 10.1515/crll.1977.296.125
- S. J. Patterson, On Dirichlet series associated with cubic Gauss sums, J. Reine Angew. Math. 303(304) (1978), 102–125. MR 514675, DOI 10.1515/crll.1978.303-304.102
- S. J. Patterson, On the distribution of Kummer sums, J. Reine Angew. Math. 303(304) (1978), 126–143. MR 514676, DOI 10.1515/crll.1978.303-304.126 144. S. J. Patterson, The distribution of general Gauss sums at prime arguments, Progress in Analytic Number Theory, vol. 2 (H. Halberstam and C. Hooley, eds.) Academic Press, New York, 1981, pp. 171-182.
- A. E. Pellet, Mémoire sur la théorie algébrique des équations, Bull. Soc. Math. France 15 (1887), 61–103 (French). MR 1504005, DOI 10.24033/bsmf.337 146. M. Schaar, Mémoire sur une formule d’analyse, Mem. Cour. Savants Etrangers, Acad. Roy. Sci. Lettres Beaux Arts Belgique 23 (1848/50), 17 pp. 147. M. Schaar, Mémoire sur la théorie des résidus quadratiques, Acad. Roy. Sci. Lettres Beaux Arts Belgique 24 (1850), 14 pp. 148. M. Schaar, Recherches sur la théorie des résidus quadratiques, Acad. Roy. Sci. Lettres Beaux Arts Belgique 25 (1850), 20 pp.
- Wolfgang M. Schmidt, Equations over finite fields. An elementary approach, Lecture Notes in Mathematics, Vol. 536, Springer-Verlag, Berlin-New York, 1976. MR 0429733, DOI 10.1007/BFb0080437 150. I. Schur, Über die Gaussschen Summen, Nachr. K. Gesell. Wiss. Göttingen, Math.-Phys. Kl., 1921, pp. 147-153.
- Issai Schur, Gesammelte Abhandlungen. Band I, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR 0462891, DOI 10.1007/978-3-642-61947-2
- Charlotte Angas Scott, The Binomial Equation x$^p$ - 1 = 0, Amer. J. Math. 8 (1886), no. 3, 261–264. MR 1507858, DOI 10.2307/2369409
- Daniel Shanks, Two theorems of Gauss, Pacific J. Math. 8 (1958), 609–612. MR 99555, DOI 10.2140/pjm.1958.8.609 154. D. Shanks, Review of C.-E. Fröberg, "Kummer’s Förmodan" (Lund University, 1973), Math. Comp. 29 (1975), 331-333.
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
- Carl Ludwig Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math. (2) 36 (1935), no. 3, 527–606 (German). MR 1503238, DOI 10.2307/1968644
- Carl Ludwig Siegel, Über die analytische Theorie der quadratischen Formen. III, Ann. of Math. (2) 38 (1937), no. 1, 212–291 (German). MR 1503335, DOI 10.2307/1968520
- Carl Ludwig Siegel, Indefinite quadratische Formen und Modulfunktionen, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, pp. 395–406 (German). MR 0022881
- Carl Ludwig Siegel, A generalization of the Epstein zeta function, J. Indian Math. Soc. (N.S.) 20 (1956), 1–10. MR 84517
- Carl Ludwig Siegel, Über das quadratische Reziprozitätsgesetz in algebraischen Zahlkörpern, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1960 (1960), 1–16 (German). MR 123553
- Carl Ludwig Siegel, Zu zwei Bemerkungen Kummers, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1964 (1964), 51–57 (German). MR 163899
- Carl Ludwig Siegel, Über die Fourierschen Koeffizienten der Eisensteinschen Reihen, Mat.-Fys. Medd. Danske Vid. Selsk. 34 (1964), no. 6, 20 pp. (1964) (German). MR 171757
- Carl Ludwig Siegel, Gesammelte Abhandlungen. Bände I, II, III, Springer-Verlag, Berlin-New York, 1966 (German). Herausgegeben von K. Chandrasekharan und H. Maass. MR 0197270 164. H. J. S. Smith, Sur l’équation a six périodes, C. R. Assoc. Franc., Reims, 1880, pp. 190-191. 165. C. L. Siegel, Report on the theory of numbers, Chelsea, New York, 1965.
- H. M. Stark, $L$-functions and character sums for quadratic forms. I, Acta Arith. 14 (1967/68), 35–50. MR 227122, DOI 10.4064/aa-14-1-35-50
- L. Stickelberger, Ueber eine Verallgemeinerung der Kreistheilung, Math. Ann. 37 (1890), no. 3, 321–367 (German). MR 1510649, DOI 10.1007/BF01721360 168. J. J. Sylvester, Sur les équations à3 et à 4 périodes des racines de l’ unité, C. R. Assoc. Franc., Reims, 1880, pp. 96-98. 169. E. D. Tabakova, A numerical investigation of Kummer cubic sums, Inst. Appl. Math. USSR Acad. Sci., Moscow, preprint no. 98, 1973, 22 pp. (Russian) 170. H. W. L. Tanner, On the binomial equation x 1 = 0: quinquisection, Proc. London Math. Soc. 18 (1887), 214-235.
- Liang Chi Ts’ao, Exponential sums over finite simple Jordan algebras and finite simple associative algebras, Duke Math. J. 42 (1975), 333–345. MR 364128
- Liang Chi Tsao, The rationality of the Fourier coefficients of certain Eisenstein series on tube domains, Compositio Math. 32 (1976), no. 3, 225–291. MR 419367
- Robert-C. Vaughan, Sommes trigonométriques sur les nombres premiers, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 16, A981–A983 (French, with English summary). MR 498434
- A. I. Vinogradov, On the cubic Gaussian sum, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 123–148 (Russian). MR 0207666
- William C. Waterhouse, The sign of the Gaussian sum, J. Number Theory 2 (1970), 363. MR 269585, DOI 10.1016/0022-314X(70)90065-X
- Heinrich Weber, Über Abel’s Summation endlicher Differenzenreihen, Acta Math. 27 (1903), no. 1, 225–233 (German). MR 1554984, DOI 10.1007/BF02421308 177. H. Weber, Lehrbuch der Algebra, Erster Band, zweite Auf., Friedrich Vieweg und Sohn, Braunschweig, 1898.
- André Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508. MR 29393, DOI 10.1090/S0002-9904-1949-09219-4
- André Weil, Jacobi sums as “Grössencharaktere”, Trans. Amer. Math. Soc. 73 (1952), 487–495. MR 51263, DOI 10.1090/S0002-9947-1952-0051263-0
- André Weil, La cyclotomie jadis et naguère, Enseign. Math. (2) 20 (1974), 247–263 (French). MR 441831
- André Weil, Sommes de Jacobi et caractères de Hecke, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1 (1974), 1–14 (French). MR 392859
- Koichi Yamamoto, On Gaussian sums with biquadratic residue characters, J. Reine Angew. Math. 219 (1965), 200–213. MR 184919, DOI 10.1515/crll.1965.219.200
- L. Carlitz, Explicit evaluation of certain exponential sums, Math. Scand. 44 (1979), no. 1, 5–16. MR 544577, DOI 10.7146/math.scand.a-11793
- Martin Eichler, Quadratische Formen und orthogonale Gruppen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Band LXIII, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1952 (German). MR 0051875, DOI 10.1007/978-3-662-01212-3
- Ronald Jacobowitz, Gauss sums and the local classification of hermitian forms, Amer. J. Math. 90 (1968), 528–552. MR 227138, DOI 10.2307/2373543
- Henri Joris, On the evaluation of Gaussian sums for non-primitive Dirichlet characters, Enseign. Math. (2) 23 (1977), no. 1-2, 13–18. MR 441888
- O. T. O’Meara, Local characterization of integral quadratic forms by Gauss sums, Amer. J. Math. 79 (1957), 687–709. MR 89869, DOI 10.2307/2372570
- James H. McClellan and Thomas W. Parks, Eigenvalue and eigenvector decomposition of the discrete Fourier transform, IEEE Trans. Audio Electroacoust. AU-20 (1972), no. 1, 66–74. MR 0399751, DOI 10.1109/TAU.1972.1162342
- Patrick Morton, On the eigenvectors of Schur’s matrix, J. Number Theory 12 (1980), no. 1, 122–127. MR 566879, DOI 10.1016/0022-314X(80)90083-9
- C.-G. Schmidt, Über die Führer von Gaußschen Summen als Größencharaktere, J. Number Theory 12 (1980), no. 3, 283–310 (German, with English summary). MR 586458, DOI 10.1016/0022-314X(80)90022-0
- Robert A. Smith, On $n$-dimensional Kloosterman sums, C. R. Math. Rep. Acad. Sci. Canada 1 (1978/79), no. 3, 173–176. MR 542305
Additional Information
- Journal: Bull. Amer. Math. Soc. 5 (1981), 107-129
- MSC (1980): Primary 10G05
- DOI: https://doi.org/10.1090/S0273-0979-1981-14930-2
- MathSciNet review: 621882