×

The determination of Gauss sums. (English) Zbl 0471.10028


MSC:

11L03 Trigonometric and exponential sums (general theory)
11R18 Cyclotomic extensions
11-02 Research exposition (monographs, survey articles) pertaining to number theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chungming An, On values of exponential sums, Proc. Amer. Math. Soc. 52 (1975), 131 – 135. · Zbl 0311.10037
[2] George E. Andrews, The theory of partitions, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. · Zbl 0371.10001
[3] Tom M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. · Zbl 0335.10001
[4] 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. · Zbl 0475.42014
[5] Raymond Ayoub, An introduction to the analytic theory of numbers, Mathematical Surveys, No. 10, American Mathematical Society, Providence, R.I., 1963. · Zbl 0128.04303
[6] P. Bachmann, Die Lehre von der Kreistheilung, Teubner, Leipzig, 1872. · JFM 04.0078.01
[7] R. P. Bambah and S. Chowla, On the sign of the Gaussian sum, Proc. Nat. Inst. Sci. India 13 (1947), 175 – 176.
[8] Klaus Barner, Zur Reziprozität quadratischer Charaktersummen in algebraischen Zahlkörpern, Monatsh. Math. 71 (1967), 369 – 384 (German). · Zbl 0153.07803 · doi:10.1007/BF01300643
[9] L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Information and Control 20 (1972), 158 – 175. · Zbl 0239.94007
[10] Richard Bellman, A brief introduction to theta functions, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1961. · Zbl 0098.28301
[11] Bruce C. Berndt, On Gaussian sums and other exponential sums with periodic coefficients, Duke Math. J. 40 (1973), 145 – 156. · Zbl 0255.10042
[12] Bruce C. Berndt and S. Chowla, The reckoning of certain quartic and octic Gauss sums, Glasgow Math. J. 18 (1977), no. 2, 153 – 155. · Zbl 0356.10032 · doi:10.1017/S0017089500003190
[13] 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. · Zbl 0412.10027 · doi:10.1016/0022-314X(79)90008-8
[14] Bruce C. Berndt and Ronald J. Evans, Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer, Illinois J. Math. 23 (1979), no. 3, 374 – 437. · Zbl 0393.12029
[15] Gudrun Beyer, Über eine Klasseneinteilung aller kubischen Restcharaktere, Abh. Math. Sem. Univ. Hamburg 19 (1954), no. 1-2, 115 – 116 (German). · Zbl 0055.27302 · doi:10.1007/BF02941559
[16] A. I. Borevich and I. R. Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. · Zbl 0145.04902
[17] Maurizio Boyarsky, \?-adic gamma functions and Dwork cohomology, Trans. Amer. Math. Soc. 257 (1980), no. 2, 359 – 369. · Zbl 0475.14017
[18] D. M. Bressoud, Applications of Andrews’ basic Lauricella transformation, Proc. Amer. Math. Soc. 72 (1978), no. 1, 89 – 94. · Zbl 0391.33004
[19] D. M. Bressoud, On the value of Gaussian sums, J. Number Theory 13 (1981), no. 1, 88 – 94. · Zbl 0443.10026 · doi:10.1016/0022-314X(81)90030-5
[20] W. Burnside, On cyclotomic quinquisection, Proc. London Math. Soc. 14 (1915), 251-259. · JFM 45.1252.01
[21] F. S. Carey, Notes on the division of the circle, Quart. J. Math. 26 (1893), 322-371. · JFM 25.0287.03
[22] L. Carlitz, A note on Gauss’ sum, Proc. Amer. Math. Soc. 7 (1956), 910 – 911. · Zbl 0072.26603
[23] L. Carlitz, A note on Gauss’s sum, Matematiche (Catania) 23 (1968), 147 – 150. · Zbl 0167.04002
[24] J. W. S. Cassels, On the determination of generalized Gauss sums, Arch. Math. (Brno) 5 (1969), 79 – 84. · Zbl 0233.10018
[25] J. W. S. Cassels, On Kummer sums, Proc. London Math. Soc. (3) 21 (1970), 19 – 27. · Zbl 0197.32004 · 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.
[27] 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.
[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. · JFM 16.0025.04
[31] A. Cayley, On the binomial equation x 1 = 0; trisection and quartisection, Proc. London Math. Soc. 11 (1879), 4-17. · JFM 12.0128.02
[32] A. Cauchy, The binomial equation x 1 = 0; quinquisection, Proc. London Math. Soc. 12 (1880), 15-16. · JFM 13.0138.05
[33] A. Cauchy, The binomial equation x 1 = 0; quinquisection, second note, Proc. London Math. Soc. 16 (1885), 61-63. · JFM 17.0153.04
[34] K. Chandrasekharan, Introduction to analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 148, Springer-Verlag New York Inc., New York, 1968. · Zbl 0169.37502
[35] 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. · Zbl 0136.32702
[36] J. G. van der Corput, Zahlentheoretische Abschätzungen, Math. Ann. 84 (1921), 53-79. · JFM 48.0181.02
[37] Harold Davenport, Multiplicative number theory, Lectures given at the University of Michigan, Winter Term, vol. 1966, Markham Publishing Co., Chicago, Ill., 1967. · Zbl 0159.06303
[38] H. Davenport and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math. 172 (1934), 151-182. · JFM 60.0913.01
[39] R. Dedekind, Review of P. Bachmann, ”Die Lehre von der Kreistheilung”, Z. Math. Phys. 18 (1873), 14-24 (in Literaturzeitung).
[40] P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin-New York, 1977. Séminaire de Géométrie Algébrique du Bois-Marie SGA 41\over2; Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier.
[41] L. E. Dickson, Cyclotomy, Higher Congruences, and Waring’s Problem, Amer. J. Math. 57 (1935), no. 2, 391 – 424. · Zbl 0012.01203 · doi:10.2307/2371217
[42] L. E. Dickson, H. H. Mitchell, H. S. Vandiver, and G. E. Wahlin, Algebraic numbers, Chelsea Publishing Co., New York, 1967. · JFM 49.0109.01
[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. · ERAM 017.0557cj
[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. · ERAM 021.0663cj
[46] P. G. L. Dirichlet, Werke, Erster Band, Georg Reimer, Berlin, 1889. · JFM 28.0014.01
[47] P. G. L. Dirichlet, Vorlesungen über Zahlentheorie, 4th. ed., Friedrich Vieweg und Sohn, Braunschweig, 1894. · JFM 03.0063.01
[48] Martin Eichler, Introduction to the theory of algebraic numbers and functions, Translated from the German by George Striker. Pure and Applied Mathematics, Vol. 23, Academic Press, New York-London, 1966. · Zbl 0152.19502
[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. · Zbl 0339.01018
[51] T. Estermann, On the sign of the Gaussian sum, J. London Math. Soc. 20 (1945), 66 – 67. · Zbl 0060.10901 · doi:10.1112/jlms/s1-20.2.66
[52] Ronald J. Evans, Generalizations of a theorem of Chowla on Gaussian sums, Houston J. Math. 3 (1977), no. 3, 343 – 349. · Zbl 0372.10028
[53] Ronald J. Evans, Unambiguous evaluations of bidecic Jacobi and Jacobsthal sums, J. Austral. Math. Soc. Ser. A 28 (1979), no. 2, 235 – 240. · Zbl 0417.10034
[54] Ronald J. Evans, The 2^{\?}th power character of 2, J. Reine Angew. Math. 315 (1980), 174 – 189. · Zbl 0419.10003 · doi:10.1515/crll.1980.315.174
[55] Ronald J. Evans, Bioctic Gauss sums and sixteenth power residue difference sets, Acta Arith. 38 (1980/81), no. 1, 37 – 46. · Zbl 0431.10022
[56] Ronald J. Evans, Twenty-fourth power residue difference sets, Math. Comp. 40 (1983), no. 162, 677 – 683. · Zbl 0513.10004
[57] Ronald J. Evans, Rational reciprocity laws, Acta Arith. 39 (1981), no. 3, 281 – 294. · Zbl 0472.10006
[58] Ronald J. Evans, Pure Gauss sums over finite fields, Mathematika 28 (1981), no. 2, 239 – 248 (1982). · Zbl 0475.10032 · doi:10.1112/S0025579300010299
[59] Ronald J. Evans, Identities for products of Gauss sums over finite fields, Enseign. Math. (2) 27 (1981), no. 3-4, 197 – 209 (1982). · Zbl 0491.12020
[60] R. Fricke, Lehrbuch der Algebra, Band I, Friedrich Vieweg und Sohn, Braunschweig, 1924. · JFM 50.0042.03
[61] Carl-Erik Fröberg, New results on the Kummer conjecture, Nordisk Tidskr. Informationsbehandling (BIT) 14 (1974), 117 – 119. · Zbl 0274.12003
[62] A. Fröhlich, Non-Abelian Jacobi sums, Number theory and algebra, Academic Press, New York, 1977, pp. 71 – 75. · Zbl 0374.12004
[63] Carl Friedrich Gauss, Mathematisches Tagebuch, 1796 – 1814, 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; Ostwalds Klassiker der Exakten Wissenschaften, Band 256.
[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. · JFM 50.0001.01
[67] Carl Friedrich Gauss, Disquisitiones arithmeticae, Translated into English by Arthur A. Clarke, S. J, Yale University Press, New Haven, Conn.-London, 1966. · Zbl 0136.32301
[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.
[70] J. C. Glashan, Quinquisection of the Cylotomic Equation, Amer. J. Math. 21 (1899), no. 3, 270 – 275. · JFM 30.0105.01 · doi:10.2307/2369604
[71] G. Gras, Sommes de Gauss sur les corps finis, Publ. Math. Besancon, 1977-78, 71 pp. · Zbl 0472.12017
[72] Benedict H. Gross and Neal Koblitz, Gauss sums and the \?-adic \Gamma -function, Ann. of Math. (2) 109 (1979), no. 3, 569 – 581. · Zbl 0406.12010 · doi:10.2307/1971226
[73] Helmut Hasse, Zahlentheorie, Dritte berichtigte Auflage, Akademie-Verlag, Berlin, 1969 (German). · Zbl 0035.02002
[74] D. R. Heath-Brown and S. J. Patterson, The distribution of Kummer sums at prime arguments, J. Reine Angew. Math. 310 (1979), 111 – 130. · Zbl 0412.10028
[75] E. Hecke, Reziprozitätsgesetz und Gausssche Summen in quadratischen Zahlkörpern, Nachr. Gesell. Wiss. Göttingen, Math.-Phys. Kl. 1919, 265-278. · JFM 47.0145.01
[76] 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. · Zbl 0092.00102
[77] E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, Chelsea, New York, 1948. · Zbl 0041.01102
[78] 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). · Zbl 0373.12009
[79] 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.
[80] 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. · Zbl 0248.10002
[81] Sonoko Ishimura, On Gaussian sums associated with a character of order 5 and a rational prime number \?\equiv 1(\?\?\?5), J. Tsuda College 8 (1976), 27 – 35.
[82] Jean-René Joly, Équations et variétés algébriques sur un corps fini, Enseignement Math. (2) 19 (1973), 1 – 117 (French). · Zbl 0282.14005
[83] A. Krazer, Lehrbuch der Thetafunktionen, Teubner, Leipzig, 1903. · JFM 34.0492.08
[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. · JFM 12.0123.01
[86] L. Kronecker, Summirung der Gaussschen Reihen \(\sum_{h=0}^{h=n 1}\;e%{\frac{2h^2\pi i}n}, J. Reine Angew. Math. 105 (1889), 267-268.\) · JFM 21.0251.01
[87] L. Kronecker, Über die Dirichletsche Methode der Wertbestimmung der Gaussschen Reihen, Mitth. Math. Gesell. Hamburg 2 (1890), 32-36. · JFM 22.0205.04
[88] Daniel S. Kubert and Serge Lang, Independence of modular units on Tate curves, Math. Ann. 240 (1979), no. 3, 191 – 210. · Zbl 0382.14008 · doi:10.1007/BF01362309
[89] Tomio Kubota, An application of the power residue theory to some abelian functions, Nagoya Math. J. 27 (1966), 51 – 54. · Zbl 0168.29601
[90] Tomio Kubota, On a special kind of Dirichlet series, J. Math. Soc. Japan 20 (1968), 193 – 207. · Zbl 0157.10202 · doi:10.2969/jmsj/02010193
[91] Tomio Kubota, Some results concerning reciprocity and functional analysis, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 395 – 399. · Zbl 0275.12006
[92] 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.
[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. · ERAM 032.0930cj
[95] E. E. Kummer, Theorie der idealen Primfaktoren der complexen Zahlen, welche aus den Wurzeln der Gleichung \omega n = 1 gebildet sind, wenn n eine zusammengesetzte Zahl ist, Math. Abh. K. Akad. Wiss. Berlin (1856), 1-47.
[96] 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. Ernst Eduard Kummer, Collected papers, Springer-Verlag, Berlin-New York, 1975. Volume II: Function theory, geometry and miscellaneous; Edited and with a foreword by André Weil.
[97] E. Landau, Über das Vorzeichen der Gaussschen Summe, Nachr. Gesell. Wiss. Göttingen, Math.-Phys. KI. 1928, 19-20. · JFM 54.0197.02
[98] Edmund Landau, Elementary number theory, Chelsea Publishing Co., New York, N.Y., 1958. Translated by J. E. Goodman. · Zbl 0079.06201
[99] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. · Zbl 0211.38404
[100] Serge Lang, Cyclotomic fields, Springer-Verlag, New York-Heidelberg, 1978. Graduate Texts in Mathematics, Vol. 59. · Zbl 0395.12005
[101] Serge Lang, Cyclotomic fields, Springer-Verlag, New York-Heidelberg, 1978. Graduate Texts in Mathematics, Vol. 59. · Zbl 0395.12005
[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. · JFM 30.0201.03
[106] Emma Lehmer, The quintic character of 2 and 3, Duke Math. J. 18 (1951), 11 – 18. · Zbl 0045.02002
[107] Emma Lehmer, On the location of Gauss sums, Math. Tables Aids Comput. 10 (1956), 194 – 202. · Zbl 0073.03001
[108] Emma Lehmer and Leonard Carlitz, Advanced Problems and Solutions: Solutions: 4636, Amer. Math. Monthly 63 (1956), no. 8, 584 – 587. · doi:10.2307/2310217
[109] G. Libri, Mémoire sur la théorie des nombres, J. Reine Angew. Math. 9 (1832), 169-188. · ERAM 009.0339cj
[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. · JFM 37.0447.09
[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.
[114] J. H. Loxton, Products related to Gauss sums, J. Reine Angew. Math. 268/269 (1974), 53 – 67. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II. · Zbl 0293.10019 · doi:10.1515/crll.1974.268-269.53
[115] 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. · Zbl 0373.10023
[116] J. H. Loxton, Some conjectures concerning Gauss sums, J. Reine Angew. Math. 297 (1978), 153 – 158. · Zbl 0362.10033 · doi:10.1515/crll.1978.297.153
[117] J. Martinet, Character theory and Artin \?-functions, Algebraic number fields: \?-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 1 – 87.
[118] G. B. Mathews, Theory of numbers, second ed., Chelsea, New York. · JFM 24.0162.01
[119] C. R. Matthews, Gauss sums and elliptic functions, Ph. D. thesis, Cambridge, 1978.
[120] C. R. Matthews, Gauss sums and elliptic functions. I. The Kummer sum, Invent. Math. 52 (1979), no. 2, 163 – 185. · Zbl 0388.10025 · doi:10.1007/BF01403063
[121] C. R. Matthews, Gauss sums and elliptic functions. II. The quartic sum, Invent. Math. 54 (1979), no. 1, 23 – 52. · Zbl 0414.10036 · doi:10.1007/BF01391175
[122] R. J. McEliece, Irreducible cyclic codes and Gauss sums, Combinatorics (Proc. NATO Advanced Study Inst., Breukelen, 1974) Math. Centrum, Amsterdam, 1974, pp. 179 – 196. Math. Centre Tracts, No. 55. · Zbl 0309.94022
[123] Andrew D. McGettrick, A result in the theory of Weierstrass elliptic functions, Proc. London Math. Soc. (3) 25 (1972), 41 – 54. · Zbl 0251.10027 · doi:10.1112/plms/s3-25.1.41
[124] Andrew D. McGettrick, On the biquadratic Gauss sum, Proc. Cambridge Philos. Soc. 71 (1972), 79 – 83. · Zbl 0226.10042
[125] F. Mertens, Ueber die Gaussischen Summen, Sitz. Berliner Akad., 1896, 217-219. · JFM 27.0144.01
[126] Howard H. Mitchell, On the generalized Jacobi-Kummer cyclotomic function, Trans. Amer. Math. Soc. 17 (1916), no. 2, 165 – 177. · JFM 46.0255.06
[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 \[ \int{\l}imits_ {-\infty}^ \infty {\tfrac{{e^ {ax^ 2 + bx} }}{{e^ ax + d}}da} \] and the analytic theory of numbers, Acta Math. 61 (1933), 323-360. · JFM 59.0197.01
[129] L. J. Mordell, The sign of the Gaussian sum, Illinois J. Math. 6 (1962), 177 – 180. · Zbl 0101.28001
[130] Carlos Julio Moreno, Sur le problème de Kummer, Enseignement Math. (2) 20 (1974), 45 – 51 (French). · Zbl 0307.12004
[131] Joseph B. Muskat and Albert L. Whiteman, The cyclotomic numbers of order twenty, Acta Arith. 17 (1970), 185 – 216. · Zbl 0216.30801
[132] Gerald Myerson, Period polynomials and Gauss sums for finite fields, Acta Arith. 39 (1981), no. 3, 251 – 264. · Zbl 0393.12028
[133] Trygve Nagell, Introduction to number theory, Second edition, Chelsea Publishing Co., New York, 1964. · Zbl 0221.10002
[134] Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, PWN — Polish Scientific Publishers, Warsaw, 1974. Monografie Matematyczne, Tom 57. · Zbl 0276.12002
[135] J. von Neumann and H. H. Goldstine, A numerical study of a conjecture of Kummer, Math. Tables and Other Aids to Computation 7 (1953), 133 – 134. · Zbl 0051.28101
[136] H. Niederreiter, On the cycle structure of linear recurring sequences, Math. Scand. 38 (1976), no. 1, 53 – 77. · Zbl 0325.12007 · doi:10.7146/math.scand.a-11616
[137] Harald Niederreiter, Weights of cyclic codes, Information and Control 34 (1977), no. 2, 130 – 140. · Zbl 0357.94008
[138] Harald Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), no. 6, 957 – 1041. · Zbl 0404.65003
[140] S. J. Patterson, A cubic analogue of the theta series, J. Reine Angew. Math. 296 (1977), 125 – 161. , https://doi.org/10.1515/crll.1977.296.125 S. J. Patterson, A cubic analogue of the theta series. II, J. Reine Angew. Math. 296 (1977), 217 – 220. · Zbl 0358.10012 · doi:10.1515/crll.1977.296.217
[141] S. J. Patterson, A cubic analogue of the theta series, J. Reine Angew. Math. 296 (1977), 125 – 161. , https://doi.org/10.1515/crll.1977.296.125 S. J. Patterson, A cubic analogue of the theta series. II, J. Reine Angew. Math. 296 (1977), 217 – 220. · Zbl 0358.10012 · doi:10.1515/crll.1977.296.217
[142] S. J. Patterson, On Dirichlet series associated with cubic Gauss sums, J. Reine Angew. Math. 303/304 (1978), 102 – 125. · Zbl 0384.10016 · doi:10.1515/crll.1978.303-304.102
[143] S. J. Patterson, On the distribution of Kummer sums, J. Reine Angew. Math. 303/304 (1978), 126 – 143. · Zbl 0384.10017 · 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 Academic Press, New York, 1981, pp. 171-182. · Zbl 0463.10027
[145] A. E. Pellet, Mémoire sur la théorie algébrique des équations, Bull. Soc. Math. France 15 (1887), 61 – 103 (French). · JFM 19.0069.01
[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.
[149] Wolfgang M. Schmidt, Equations over finite fields. An elementary approach, Lecture Notes in Mathematics, Vol. 536, Springer-Verlag, Berlin-New York, 1976. · Zbl 0329.12001
[150] I. Schur, Über die Gaussschen Summen, Nachr. K. Gesell. Wiss. Göttingen, Math.-Phys. Kl., 1921, pp. 147-153. · JFM 48.0130.07
[151] Issai Schur, Gesammelte Abhandlungen. Band I, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. Issai Schur, Gesammelte Abhandlungen. Band II, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. Issai Schur, Gesammelte Abhandlungen. Band III, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. · Zbl 0274.01054
[152] Charlotte Angas Scott, The Binomial Equation x^{\?} - 1 = 0, Amer. J. Math. 8 (1886), no. 3, 261 – 264. · JFM 18.0066.03 · doi:10.2307/2369409
[153] Daniel Shanks, Two theorems of Gauss, Pacific J. Math. 8 (1958), 609 – 612. · Zbl 0084.06003
[154] D. Shanks, Review of C.-E. Fröberg, ”Kummer’s Förmodan” (Lund University, 1973), Math. Comp. 29 (1975), 331-333.
[155] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. · Zbl 0221.10029
[156] Carl Ludwig Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math. (2) 36 (1935), no. 3, 527 – 606 (German). · Zbl 0012.19703 · doi:10.2307/1968644
[157] Carl Ludwig Siegel, Über die analytische Theorie der quadratischen Formen. III, Ann. of Math. (2) 38 (1937), no. 1, 212 – 291 (German). · Zbl 0016.01205 · doi:10.2307/1968520
[158] Carl Ludwig Siegel, Indefinite quadratische Formen und Modulfunktionen, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publisher, Inc., New York, 1948, pp. 395 – 406 (German). · Zbl 0033.01304
[159] Carl Ludwig Siegel, A generalization of the Epstein zeta function, J. Indian Math. Soc. (N.S.) 20 (1956), 1 – 10. · Zbl 0075.25103
[160] Carl Ludwig Siegel, Über das quadratische Reziprozitätsgesetz in algebraischen Zahlkörpern, Nachr. Akad. Wiss. Göttingen math.-Phys. Kl. II 1960 (1960), 1 – 16 (German). · Zbl 0109.26604
[161] Carl Ludwig Siegel, Zu zwei Bemerkungen Kummers, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1964 (1964), 51 – 57 (German). · Zbl 0119.27701
[162] Carl Ludwig Siegel, Über die Fourierschen Koeffizienten der Eisensteinschen Reihen, Mat.-Fys. Medd. Danske Vid. Selsk. 34 (1964), no. 6, 20 pp. (1964) (German). · Zbl 0132.06401
[163] Carl Ludwig Siegel, Gesammelte Abhandlungen. Bände I, II, III, Herausgegeben von K. Chandrasekharan und H. Maass, Springer-Verlag, Berlin-New York, 1966 (German). · Zbl 0143.00101
[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.
[166] H. M. Stark, \?-functions and character sums for quadratic forms. I, Acta Arith. 14 (1967/1968), 35 – 50. · Zbl 0198.37801
[167] L. Stickelberger, Ueber eine Verallgemeinerung der Kreistheilung, Math. Ann. 37 (1890), no. 3, 321 – 367 (German). · JFM 22.0100.01 · 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. · JFM 19.0085.01
[171] Liang Chi Ts’ao, Exponential sums over finite simple Jordan algebras and finite simple associative algebras, Duke Math. J. 42 (1975), 333 – 345. · Zbl 0348.10027
[172] Liang Chi Tsao, The rationality of the Fourier coefficients of certain Eisenstein series on tube domains, Compositio Math. 32 (1976), no. 3, 225 – 291. · Zbl 0346.10014
[173] 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). · Zbl 0374.10025
[174] A. I. Vinogradov, On the cubic Gaussian sum, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 123 – 148 (Russian). · Zbl 0157.08903
[175] William C. Waterhouse, The sign of the Gaussian sum, J. Number Theory 2 (1970), 363. · Zbl 0197.32003 · doi:10.1016/0022-314X(70)90065-X
[176] Heinrich Weber, Über Abel’s Summation endlicher Differenzenreihen, Acta Math. 27 (1903), no. 1, 225 – 233 (German). · JFM 34.0376.01 · doi:10.1007/BF02421308
[177] H. Weber, Lehrbuch der Algebra, Erster Band, zweite Auf., Friedrich Vieweg und Sohn, Braunschweig, 1898. · JFM 29.0064.01
[178] André Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497 – 508. · Zbl 0032.39402
[179] André Weil, Jacobi sums as ”Grössencharaktere”, Trans. Amer. Math. Soc. 73 (1952), 487 – 495. · Zbl 0048.27001
[180] André Weil, La cyclotomie jadis et naguère, Enseignement Math. (2) 20 (1974), 247 – 263 (French). · Zbl 0352.12006
[181] André Weil, Sommes de Jacobi et caractères de Hecke, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1 (1974), 1 – 14 (French). · Zbl 0367.10035
[182] Koichi Yamamoto, On Gaussian sums with biquadratic residue characters, J. Reine Angew. Math. 219 (1965), 200 – 213. · Zbl 0133.29306 · doi:10.1515/crll.1965.219.200
[183] L. Carlitz, Explicit evaluation of certain exponential sums, Math. Scand. 44 (1979), no. 1, 5 – 16. · Zbl 0396.12017 · doi:10.7146/math.scand.a-11793
[184] 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). · Zbl 0049.31106
[185] Ronald Jacobowitz, Gauss sums and the local classification of hermitian forms, Amer. J. Math. 90 (1968), 528 – 552. · Zbl 0176.33702 · doi:10.2307/2373543
[186] Henri Joris, On the evaluation of Gaussian sums for non-primitive Dirichlet characters, Enseignement Math. (2) 23 (1977), no. 1 – 2, 13 – 18. · Zbl 0352.10018
[187] O. T. O’Meara, Local characterization of integral quadratic forms by Gauss sums, Amer. J. Math. 79 (1957), 687 – 709. · Zbl 0078.02904 · doi:10.2307/2372570
[188] 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.
[189] Patrick Morton, On the eigenvectors of Schur’s matrix, J. Number Theory 12 (1980), no. 1, 122 – 127. · Zbl 0428.10020 · doi:10.1016/0022-314X(80)90083-9
[190] C.-G. Schmidt, Über die Führer von Gaußschen Summen als Größencharaktere, J. Number Theory 12 (1980), no. 3, 283 – 310 (German, with English summary). · Zbl 0446.12002 · doi:10.1016/0022-314X(80)90022-0
[191] Robert A. Smith, On \?-dimensional Kloosterman sums, C. R. Math. Rep. Acad. Sci. Canada 1 (1978/79), no. 3, 173 – 176.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.