메뉴 건너뛰기




Volumn 29, Issue 3, 2002, Pages 358-385

Explicit evaluations and reciprocity theorems for finite trigonometric sums

Author keywords

[No Author keywords available]

Indexed keywords


EID: 0036821845     PISSN: 01968858     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0196-8858(02)00020-9     Document Type: Article
Times cited : (91)

References (89)
  • 1
    • 0012569576 scopus 로고
    • Problem 4420: A summation of tangents
    • Solutions by J.B. Rosser and the proposer, and F. Underwood
    • F.S. Acton, Problem 4420: A summation of tangents, Solutions by J.B. Rosser and the proposer, and F. Underwood, Amer. Math. Monthly 59 (1952) 337-338.
    • (1952) Amer. Math. Monthly , vol.59 , pp. 337-338
    • Acton, F.S.1
  • 2
    • 0032325342 scopus 로고    scopus 로고
    • Asymptotic formulas and generalized Dedekind sums
    • G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Experiment. Math. 7 (1998) 343-359.
    • (1998) Experiment. Math. , vol.7 , pp. 343-359
    • Almkvist, G.1
  • 3
    • 84972533487 scopus 로고
    • Theorems on generalized Dedekind sums
    • T.M. Apostol, Theorems on generalized Dedekind sums, Pacific J. Math. 2 (1952) 1-9.
    • (1952) Pacific J. Math. , vol.2 , pp. 1-9
    • Apostol, T.M.1
  • 4
    • 0012518385 scopus 로고
    • Another elementary proof of Euler's formula for ζ(2n)
    • T.M. Apostol, Another elementary proof of Euler's formula for ζ(2n), Amer. Math. Monthly 80 (1973) 425-431.
    • (1973) Amer. Math. Monthly , vol.80 , pp. 425-431
    • Apostol, T.M.1
  • 5
    • 0012575866 scopus 로고
    • Problem 879 Solutions by J. Boersma and A.A. Jagers
    • S.-J. Bang, Problem 879, Solutions by J. Boersma and A.A. Jagers, Nieuw Arch. Wiss. 12 (1994) 106-108.
    • (1994) Nieuw Arch. Wiss. , vol.12 , pp. 106-108
    • Bang, S.-J.1
  • 6
    • 0033442825 scopus 로고    scopus 로고
    • The reciprocity law for Dedekind sums via the constant Ehrhart coefficient
    • M. Beck, The reciprocity law for Dedekind sums via the constant Ehrhart coefficient, Amer. Math. Monthly 106 (1999) 459-462.
    • (1999) Amer. Math. Monthly , vol.106 , pp. 459-462
    • Beck, M.1
  • 7
    • 4243551106 scopus 로고    scopus 로고
    • Dedekind cotangent sums
    • in preparation
    • M. Beck, Dedekind cotangent sums, in preparation.
    • Beck, M.1
  • 9
    • 0012607441 scopus 로고
    • Aufgabe 828 Solution by A.A. Jagers
    • M. Bencze, Aufgabe 828, Solution by A.A. Jagers, Elem. Math. 35 (1980) 123.
    • (1980) Elem. Math. , vol.35 , pp. 123
    • Bencze, M.1
  • 10
    • 0012569578 scopus 로고
    • Sommen van kwadraten van cotangenten
    • M. Bencze, Sommen van kwadraten van cotangenten, Nieuw Tijdschr. Wisk. 67 (1980) 141-144.
    • (1980) Nieuw Tijdschr. Wisk. , vol.67 , pp. 141-144
    • Bencze, M.1
  • 11
    • 0012617740 scopus 로고
    • Elementary evaluation of ζ(2n)
    • B.C. Berndt, Elementary evaluation of ζ(2n), Math. Magazine 48 (1975) 148-154.
    • (1975) Math. Magazine , vol.48 , pp. 148-154
    • Berndt, B.C.1
  • 12
    • 84963085208 scopus 로고
    • Dedekind sums and a paper of G.H. Hardy
    • B.C. Berndt, Dedekind sums and a paper of G.H. Hardy, J. London Math. Soc. (2) 13 (1976) 129-137.
    • (1976) J. London Math. Soc. , vol.13 , Issue.2 , pp. 129-137
    • Berndt, B.C.1
  • 13
    • 0000480746 scopus 로고
    • Reciprocity theorems for Dedekind sums and generalizations
    • B.C. Berndt, Reciprocity theorems for Dedekind sums and generalizations. Adv. Math. 23 (1977) 285-316.
    • (1977) Adv. Math. , vol.23 , pp. 285-316
    • Berndt, B.C.1
  • 14
    • 0000546814 scopus 로고
    • Analytic properties of arithmetic sums arising in the theory of the classical theta-functions
    • B.C. Berndt, L.A. Goldberg, Analytic properties of arithmetic sums arising in the theory of the classical theta-functions, SIAM J. Math. Anal. 15 (1984) 143-150.
    • (1984) SIAM J. Math. Anal. , vol.15 , pp. 143-150
    • Berndt, B.C.1    Goldberg, L.A.2
  • 15
    • 4244106534 scopus 로고
    • Problem 6332: Asymptotic formulas from Chebyshev polynomials
    • Solution by H.-J
    • P.E. Bjørstad, H.E. Fettis, Problem 6332: Asymptotic formulas from Chebyshev polynomials, Solution by H.-J. Seiffert. Amer. Math. Monthly 101 (1994) 1015-1017.
    • (1994) Seiffert. Amer. Math. Monthly , vol.101 , pp. 1015-1017
    • Bjørstad, P.E.1    Fettis, H.E.2
  • 16
    • 14244263756 scopus 로고
    • Beispiele zur Differentialtopologie von Singularitäten
    • E. Breiskorn, Beispiele zur Differentialtopologie von Singularitäten, Invent. Math. 2 (1966) 1-14.
    • (1966) Invent. Math. , vol.2 , pp. 1-14
    • Breiskorn, E.1
  • 18
    • 0012617986 scopus 로고
    • Problem 5486: Higher trigonometric identities
    • Solutions by W.O. Egerland, W.J. Blundon, and J.H. van Lint
    • L. Bruch, Problem 5486: Higher trigonometric identities, Solutions by W.O. Egerland, W.J. Blundon, and J.H. van Lint, Amer. Math. Monthly 75 (1968) 421-422.
    • (1968) Amer. Math. Monthly , vol.75 , pp. 421-422
    • Bruch, L.1
  • 19
    • 0012567559 scopus 로고
    • Problem H-349: Triggy
    • Solution by Ö. Eǧecioǧlu
    • P.S. Bruckman, Problem H-349: Triggy, Solution by Ö. Eǧecioǧlu, Fibonacci Quart. 22 (1984) 190-191.
    • (1984) Fibonacci Quart. , vol.22 , pp. 190-191
    • Bruckman, P.S.1
  • 21
    • 0002044031 scopus 로고
    • Some diophantine relations involving circular functions of rational angles
    • F. Calogero, A.M. Perelomov, Some diophantine relations involving circular functions of rational angles, Linear Algebra Appl. 25 (1979) 91-94.
    • (1979) Linear Algebra Appl. , vol.25 , pp. 91-94
    • Calogero, F.1    Perelomov, A.M.2
  • 22
    • 84972508706 scopus 로고
    • A note on generalized Dedekind sums
    • L. Carlitz, A note on generalized Dedekind sums, Duke Math. J. 21 (1954) 399-403.
    • (1954) Duke Math. J. , vol.21 , pp. 399-403
    • Carlitz, L.1
  • 23
    • 0012611989 scopus 로고
    • Many-term relations for multiple Dedekind sums
    • L. Carlitz, Many-term relations for multiple Dedekind sums, Indian J. Math. 20 (1978) 77-89.
    • (1978) Indian J. Math. , vol.20 , pp. 77-89
    • Carlitz, L.1
  • 24
    • 0033174122 scopus 로고    scopus 로고
    • Partial fractions and trigonometric identities
    • W. Chu, A. Marini, Partial fractions and trigonometric identities, Adv. Appl. Math. 23 (1999) 115-175.
    • (1999) Adv. Appl. Math. , vol.23 , pp. 115-175
    • Chu, W.1    Marini, A.2
  • 25
    • 0012520831 scopus 로고    scopus 로고
    • Finite cotangent sums and the Riemann zeta function
    • D. Cvijović, J. Klinowski, Finite cotangent sums and the Riemann zeta function, Math. Slovaca 50 (2000) 149-157.
    • (2000) Math. Slovaca , vol.50 , pp. 149-157
    • Cvijović, D.1    Klinowski, J.2
  • 26
    • 0001058619 scopus 로고
    • Cotangent sums, a further generalization of Dedekind sums
    • U. Dieter, Cotangent sums, a further generalization of Dedekind sums, J. Number Theory 18 (1984) 289-305.
    • (1984) J. Number Theory , vol.18 , pp. 289-305
    • Dieter, U.1
  • 27
    • 0001288321 scopus 로고
    • Casimir effect around a cone
    • J.S. Dowker, Casimir effect around a cone, Phys. Rev. D 36 (1987) 3095-3101.
    • (1987) Phys. Rev. D , vol.36 , pp. 3095-3101
    • Dowker, J.S.1
  • 28
    • 0000993237 scopus 로고
    • Heat kernel expansion on a generalized cone
    • J.S. Dowker, Heat kernel expansion on a generalized cone, J. Math. Phys. 30 (1989) 770-773.
    • (1989) J. Math. Phys. , vol.30 , pp. 770-773
    • Dowker, J.S.1
  • 29
    • 0012577093 scopus 로고
    • Problem E3339: A sequence of polynomials related to the roots of unity
    • Solution by R.J. Chapman and R.W.K. Odoni
    • A.J. Duran, Problem E3339: A sequence of polynomials related to the roots of unity, Solution by R.J. Chapman and R.W.K. Odoni, Amer. Math. Monthly 98 (1991) 269-271.
    • (1991) Amer. Math. Monthly , vol.98 , pp. 269-271
    • Duran, A.J.1
  • 30
    • 0000443923 scopus 로고
    • Aufgaben und Lehrsätze
    • G. Eisenstein, Aufgaben und Lehrsätze, J. Reine Angew. Math. 27 (1844) 281-283.
    • (1844) J. Reine Angew. Math. , vol.27 , pp. 281-283
    • Eisenstein, G.1
  • 31
    • 26344434047 scopus 로고
    • Nachtrag zum cubischen Reciprocitätssatze für die aus dritten Wurzeln der Einheit zusammengesetzten complexen Zahlen. Criterien des cubischen Characters der Zahl 3 und ihrer Theiler
    • G. Eisenstein, Nachtrag zum cubischen Reciprocitätssatze für die aus dritten Wurzeln der Einheit zusammengesetzten complexen Zahlen. Criterien des cubischen Characters der Zahl 3 und ihrer Theiler, J. Reine Angew. Math. 28 (1844) 28-35.
    • (1844) J. Reine Angew. Math. , vol.28 , pp. 28-35
    • Eisenstein, G.1
  • 32
    • 0011372094 scopus 로고
    • Mathematische Werke
    • Band I, Chelsea, New York
    • G. Eisenstein, Mathematische Werke, Band I, Chelsea, New York, 1975.
    • (1975)
    • Eisenstein, G.1
  • 33
    • 0012617987 scopus 로고
    • Problem 1198: A case of Euler's identity
    • Solutions by J.C. Linders and W.A. Newcomb
    • R. Euler, Problem 1198: A case of Euler's identity, Solutions by J.C. Linders and W.A. Newcomb, Math. Magazine 58 (1985) 242-243.
    • (1985) Math. Magazine , vol.58 , pp. 242-243
    • Euler, R.1
  • 34
    • 0032345040 scopus 로고    scopus 로고
    • Modular forms, generalized Dedekind symbols and period polynomials
    • S. Fukuhara, Modular forms, generalized Dedekind symbols and period polynomials, Math. Ann. 310 (1998) 83-101.
    • (1998) Math. Ann. , vol.310 , pp. 83-101
    • Fukuhara, S.1
  • 35
    • 0012520148 scopus 로고    scopus 로고
    • New trigonometric identities and generalized Dedekind sums
    • submitted
    • S. Fukuhara, New trigonometric identities and generalized Dedekind sums, submitted.
    • Fukuhara, S.1
  • 36
    • 85064188011 scopus 로고    scopus 로고
    • Dedekind symbols associated with J-form and their reciprocity law
    • to appear
    • S. Fukuhara, Dedekind symbols associated with J-form and their reciprocity law, J. Number Theory, to appear.
    • J. Number Theory
    • Fukuhara, S.1
  • 37
    • 0012519075 scopus 로고    scopus 로고
    • Higher dimensional Dedekind sums with a complex parameter
    • Preprint
    • S. Fukuhara, Higher dimensional Dedekind sums with a complex parameter, Preprint.
    • Fukuhara, S.1
  • 38
    • 22944492848 scopus 로고    scopus 로고
    • A generalized reciprocity law and w-invariants of law spaces
    • submitted
    • Y. Fukumoto, A generalized reciprocity law and w-invariants of law spaces, submitted.
    • Fukumoto, Y.1
  • 39
    • 0012617988 scopus 로고
    • On sums analogous to Dedekind's sums
    • Proc. Fifth Manitoba Conf. on Numerical Mathematics, Congressus Numerantium, No. XVI, Utilitas Math. Inc., Winnipeg, MB
    • J.M. Gandhi, On sums analogous to Dedekind's sums, in: Proc. Fifth Manitoba Conf. on Numerical Mathematics, Congressus Numerantium, No. XVI, Utilitas Math. Inc., Winnipeg, MB, 1976, pp. 647-655.
    • (1976) , pp. 647-655
    • Gandhi, J.M.1
  • 40
    • 85009721084 scopus 로고
    • Problem 69-14, Sums of Inverse Powers of Cosines
    • Solution by M.E. Fisher
    • L.A. Gardner Jr., Problem 69-14, Sums of Inverse Powers of Cosines, Solution by M.E. Fisher, SIAM Rev. 13 (1971) 116-119.
    • (1971) SIAM Rev. , vol.13 , pp. 116-119
    • Gardner L.A., Jr.1
  • 41
    • 21844484985 scopus 로고
    • Some trigonometric identities encountered by McCoy and Orrick
    • A. Gervois, M.L. Mehta, Some trigonometric identities encountered by McCoy and Orrick, J. Math. Phys. 36 (1995) 5098-5109.
    • (1995) J. Math. Phys. , vol.36 , pp. 5098-5109
    • Gervois, A.1    Mehta, M.L.2
  • 42
    • 21344467542 scopus 로고    scopus 로고
    • Some trigonometric identities II
    • A. Gervois, M.L. Mehta, Some trigonometric identities II, J. Math. Phys. 37 (1996) 4150-4155.
    • (1996) J. Math. Phys. , vol.37 , pp. 4150-4155
    • Gervois, A.1    Mehta, M.L.2
  • 43
    • 13744262379 scopus 로고    scopus 로고
    • Generating functions and generalized Dedekind sums
    • Research paper R11
    • I. Gessel, Generating functions and generalized Dedekind sums, Electron. J. Combin. 4 (1997), Research paper, R11, 17 p.
    • (1997) Electron. J. Combin. , vol.4 , pp. 17
    • Gessel, I.1
  • 45
    • 0012574464 scopus 로고
    • Dedekind-Rademacher sums
    • E. Grosswald, Dedekind-Rademacher sums, Amer. Math. Monthly 78 (1971) 639-644.
    • (1971) Amer. Math. Monthly , vol.78 , pp. 639-644
    • Grosswald, E.1
  • 46
    • 0003682848 scopus 로고
    • A Table of Series and Products
    • Prentice Hall, Englewood Cliffs, NJ
    • E.R. Hansen, A Table of Series and Products, Prentice Hall, Englewood Cliffs, NJ, 1975.
    • (1975)
    • Hansen, E.R.1
  • 47
    • 0002475887 scopus 로고
    • On certain series of discontinuous functions connected with the modular functions
    • G.H. Hardy, On certain series of discontinuous functions connected with the modular functions, Quart. J. Math. 36 (1905) 93-123.
    • (1905) Quart. J. Math. , vol.36 , pp. 93-123
    • Hardy, G.H.1
  • 49
    • 0012519113 scopus 로고
    • Problem E2442: A sum involving roots of unity
    • Solutions by E.H. Umberger, and M.R. Murty, and V.K. Murty
    • J.C. Hemperly, Problem E2442: A sum involving roots of unity, Solutions by E.H. Umberger, and M.R. Murty, and V.K. Murty, Amer. Math. Monthly 81 (1974) 1031-1033.
    • (1974) Amer. Math. Monthly , vol.81 , pp. 1031-1033
    • Hemperly, J.C.1
  • 50
    • 0012570243 scopus 로고
    • Problem 86-5: Conjectured trigonometrical identities
    • Solution by S.W. Graham and O. Ruehr
    • M. Henkel, Problem 86-5: Conjectured trigonometrical identities, Solution by S.W. Graham and O. Ruehr, SIAM Review 29 (1987) 132-134.
    • (1987) SIAM Review , vol.29 , pp. 132-134
    • Henkel, M.1
  • 52
    • 0013001726 scopus 로고
    • The signature theorem: Reminiscences and recreation, Prospects in Mathematics
    • Princeton Univ. Press, Princeton
    • F. Hirzebruch, The signature theorem: reminiscences and recreation, Prospects in Mathematics, in: Ann. of Math. Stud., Vol. 70, Princeton Univ. Press, Princeton, 1971, pp. 3-31.
    • (1971) Ann. of Math. Stud. , vol.70 , pp. 3-31
    • Hirzebruch, F.1
  • 53
    • 0004060374 scopus 로고
    • The Atiyah-Singer Theorem and Elementary Number Theory
    • Publish or Perish, Boston
    • F. Hirzebruch, D. Zagier, The Atiyah-Singer Theorem and Elementary Number Theory, Publish or Perish, Boston, 1974.
    • (1974)
    • Hirzebruch, F.1    Zagier, D.2
  • 55
    • 0012520833 scopus 로고
    • A proof of a transformation formula in the theory of partitions
    • K. Iseki, A proof of a transformation formula in the theory of partitions, J. Math. Soc. Japan 4 (1952) 14-26.
    • (1952) J. Math. Soc. Japan , vol.4 , pp. 14-26
    • Iseki, K.1
  • 56
    • 84967537066 scopus 로고
    • An elementary proof of the generalized Eisenstein formula
    • M. Ishibashi, An elementary proof of the generalized Eisenstein formula, Österr. Akad. Wiss., Math. Naturwiss. Kl. 197 (1988) 443-447.
    • (1988) Österr. Akad. Wiss., Math. Naturwiss. Kl. , vol.197 , pp. 443-447
    • Ishibashi, M.1
  • 57
    • 0034345560 scopus 로고    scopus 로고
    • Multiple cotangent and generalized eta functions
    • M. Ishibashi, Multiple cotangent and generalized eta functions, Ramanujan J. 4 (2000) 221-229.
    • (2000) Ramanujan J. , vol.4 , pp. 221-229
    • Ishibashi, M.1
  • 58
    • 0004250435 scopus 로고
    • Summation of Series
    • Chapman & Hall, London
    • L.B.W. Jolley, Summation of Series, Chapman & Hall, London, 1925.
    • (1925)
    • Jolley, L.B.W.1
  • 59
    • 0012568919 scopus 로고
    • Problem 72-27, A triple sum
    • Solution by W.B. Jordan
    • P.G. Kirmser, K.K. Hu, Problem 72-27, A triple sum, Solution by W.B. Jordan, SIAM Rev. 15 (1973) 802-803.
    • (1973) SIAM Rev. , vol.15 , pp. 802-803
    • Kirmser, P.G.1    Hu, K.K.2
  • 60
    • 0012519114 scopus 로고
    • Proof of a conjecture of 1881 on permanents
    • R. Kittappa, Proof of a conjecture of 1881 on permanents, Linear and Multlinear Algebra 10 (1981) 75-82.
    • (1981) Linear and Multlinear Algebra , vol.10 , pp. 75-82
    • Kittappa, R.1
  • 61
    • 0012518216 scopus 로고
    • A note on trigonometric sums arising in gauge theory
    • T. Lawson, A note on trigonometric sums arising in gauge theory, Manuscripta Math. 80 (1993) 265-272.
    • (1993) Manuscripta Math. , vol.80 , pp. 265-272
    • Lawson, T.1
  • 63
    • 0030552840 scopus 로고    scopus 로고
    • Analyticity and integrability in the chiral Potts model
    • B.M. McCoy, W.P. Orrick, Analyticity and integrability in the chiral Potts model, J. Statist. Phys. 83 (1996) 839-865.
    • (1996) J. Statist. Phys. , vol.83 , pp. 839-865
    • McCoy, B.M.1    Orrick, W.P.2
  • 64
  • 65
    • 0002348463 scopus 로고
    • Über eine topologische und zahlentheoretische Anwendung von Hirzebruchs Spitzenauflösung
    • W. Meyer, R. Sczech, Über eine topologische und zahlentheoretische Anwendung von Hirzebruchs Spitzenauflösung, Math. Ann. 240 (1979) 69-96.
    • (1979) Math. Ann. , vol.240 , pp. 69-96
    • Meyer, W.1    Sczech, R.2
  • 66
    • 84972526538 scopus 로고
    • On certain sums generating the Dedekind sums and their reciprocity laws
    • M. Mikolás, On certain sums generating the Dedekind sums and their reciprocity laws, Pacific J. Math. 7 (1957) 1167-1178.
    • (1957) Pacific J. Math. , vol.7 , pp. 1167-1178
    • Mikolás, M.1
  • 67
    • 0012573574 scopus 로고
    • On a conjecture of R.F. Scott (1881)
    • H. Minc, On a conjecture of R.F. Scott (1881), Linear Algebra Appl.28 (1979) 141-153.
    • (1979) Linear Algebra Appl. , vol.28 , pp. 141-153
    • Minc, H.1
  • 70
    • 0012519799 scopus 로고
    • On an extension of a theorem of S. Chowla
    • T. Okada, On an extension of a theorem of S. Chowla, Acta Arith. 38 (1981) 341-345.
    • (1981) Acta Arith. , vol.38 , pp. 341-345
    • Okada, T.1
  • 73
    • 0002536743 scopus 로고
    • Egy Reciprocitásképletröl a Modulfüggevények Elméletéböl
    • H. Rademacher, Egy Reciprocitásképletröl a Modulfüggevények Elméletéböl, Mat. Fiz. Lapok 40 (1933) 24-34.
    • (1933) Mat. Fiz. Lapok , vol.40 , pp. 24-34
    • Rademacher, H.1
  • 74
    • 0012567562 scopus 로고
    • Collected Papers, MIT Press, Cambridge, MA
    • H. Rademacher, Collected Papers, Vol. II, MIT Press, Cambridge, MA, 1974.
    • (1974) , vol.2
    • Rademacher, H.1
  • 75
    • 0003227956 scopus 로고
    • Dedekind Sums
    • Math. Assoc. of America, Washington, DC
    • H. Rademacher, E. Grosswald, Dedekind Sums, in: Carus Math. Monograph, Vol. 16, Math. Assoc. of America, Washington, DC, 1972.
    • (1972) Carus Math. Monograph , vol.16
    • Rademacher, H.1    Grosswald, E.2
  • 76
    • 0012519077 scopus 로고
    • Problem 3741, Solution by L.S. Johnson
    • H.D. Ruderman, Problem 3741, Solution by L.S. Johnson, Amer. Math. Monthly 44 (1937) 254-256.
    • (1937) Amer. Math. Monthly , vol.44 , pp. 254-256
    • Ruderman, H.D.1
  • 77
    • 0012517372 scopus 로고
    • Cambridge Univ. Press, Cambridge
    • A.W. Siddons, R.T. Hughes, Trigonometry, Part IV, Cambridge Univ. Press, Cambridge, 1929.
    • (1929) Trigonometry , Issue.PART 4
    • Siddons, A.W.1    Hughes, R.T.2
  • 79
    • 84966235340 scopus 로고
    • Invariants of finite groups and their applications to combinatorics
    • R.P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979) 475-511.
    • (1979) Bull. Amer. Math. Soc. (N.S.) , vol.1 , pp. 475-511
    • Stanley, R.P.1
  • 81
    • 0012612685 scopus 로고
    • Ueber einige Eigenschaften der Function Ex
    • M. Stern, Ueber einige Eigenschaften der Function Ex, J. Reine Angew. Math. 59 (1861) 146-162.
    • (1861) J. Reine Angew. Math. , vol.59 , pp. 146-162
    • Stern, M.1
  • 82
    • 0003848069 scopus 로고
    • Introduction to Classical Real Analysis
    • Wadsworth. Belmont, CA
    • K.R. Stromberg, Introduction to Classical Real Analysis, Wadsworth, Belmont, CA, 1981.
    • (1981)
    • Stromberg, K.R.1
  • 83
    • 84972535630 scopus 로고
    • Conformal field theory and the cohomology of the moduli space of stable bundles
    • M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Differential Geom. 35 (1992) 131-149.
    • (1992) J. Differential Geom. , vol.35 , pp. 131-149
    • Thaddeus, M.1
  • 84
    • 84968464896 scopus 로고
    • Exponential sums of Lerch's zeta function
    • K. Wang, Exponential sums of Lerch's zeta function, Proc. Amer. Math. Soc. 95 (1985) 11-15.
    • (1985) Proc. Amer. Math. Soc. , vol.95 , pp. 11-15
    • Wang, K.1
  • 87
    • 0012574468 scopus 로고
    • Evaluation of two trigonometric sums
    • K.S. Williams, N.-Y. Zhang, Evaluation of two trigonometric sums, Math. Slovaca 44 (1994) 575-583.
    • (1994) Math. Slovaca , vol.44 , pp. 575-583
    • Williams, K.S.1    Zhang, N.-Y.2
  • 88
    • 0012520841 scopus 로고
    • Equivariant Pontrjagin Classes and Applications to Orbit Spaces
    • Springer-Verlag, Berlin
    • D. Zagier, Equivariant Pontrjagin Classes and Applications to Orbit Spaces, in: Lecture Notes in Math., Vol. 290, Springer-Verlag, Berlin, 1972.
    • (1972) Lecture Notes in Math. , vol.290
    • Zagier, D.1
  • 89
    • 0000315968 scopus 로고
    • Higher dimensional Dedekind sums
    • D. Zagier, Higher dimensional Dedekind sums, Math. Ann. 202 (1973) 149-172.
    • (1973) Math. Ann. , vol.202 , pp. 149-172
    • Zagier, D.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.