Trace formula in noncommutative Geometry and the zeros of the Riemann zeta function

Alain Connes

Journées équations aux dérivées partielles (1997)

  • page 1-28
  • ISSN: 0752-0360

How to cite


Connes, Alain. "Trace formula in noncommutative Geometry and the zeros of the Riemann zeta function." Journées équations aux dérivées partielles (1997): 1-28. <>.

author = {Connes, Alain},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-28},
publisher = {Ecole polytechnique},
title = {Trace formula in noncommutative Geometry and the zeros of the Riemann zeta function},
url = {},
year = {1997},

AU - Connes, Alain
TI - Trace formula in noncommutative Geometry and the zeros of the Riemann zeta function
JO - Journées équations aux dérivées partielles
PY - 1997
PB - Ecole polytechnique
SP - 1
EP - 28
LA - eng
UR -
ER -


  1. [AB] M.F. Atiyah and R. Bott, A Lefchetz fixed point formula for elliptic complexes: I, Annals of Math, 86 (1967), 374-407. Zbl0161.43201MR35 #3701
  2. [B] M. Berry, Riemann's zeta function: a model of quantum chaos, Lecture Notes in Physics, 263, Springer (1986). Zbl0664.10021
  3. [Bg] A. Beurling, A closure problem related to the Riemann zeta function, Proc. Nat. Ac. Sci. 41 (1955), 312-314. Zbl0065.30303MR17,15a
  4. [B-C] J.-B. Bost and A. Connes, Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Mathematica, New Series 1, No. 3 (1995), 411-457. Zbl0842.46040MR96m:46112
  5. [BG] O. Bohigas and M. Giannoni, Chaotic motion and random matrix theories, Lecture Notes in Physics, 209 (1984), 1-99. MR86c:58129
  6. [C] A. Connes, Noncommutative Geometry, Academic Press (1994). Zbl0818.46076MR95j:46063
  7. [Co] A. Connes, Formule de trace en Géométrie non commutative et hypothèse de Riemann, C.R. Acad. Sci. Paris Ser. A-B (1996). Zbl0864.46042MR97k:11124
  8. [D] C. Deninger, Local L-factors of motives and regularised determinants, Invent. Math., 107 (1992), 135-150. Zbl0762.14015MR93a:11056
  9. [G] D. Goldfeld, A spectral interpretation of Weil's explicit formula, Lecture Notes in Math., 1593, Springer Verlag (1994), 135-152. Zbl0828.11031MR96f:11110
  10. [GS] V. Guillemin and S. Sternberg, Geometric asymptotics, Math. Surveys, 14, Amer. Math. Soc., Providence, R.I. (1977). Zbl0364.53011MR58 #24404
  11. [Gu] V. Guillemin, Lectures on spectral theory of elliptic operators, Duke Math. J., 44, No.3 (1977), 485-517. Zbl0447.58033MR56 #6758
  12. [H] S. Haran, Riesz potentials and explicit sums in arithmetic, Invent. Math., 101 (1990), 697-703. Zbl0788.11055MR91g:11132
  13. [J] B. Julia, Statistical theory of numbers, Number Theory and Physics, Springer Proceedings in Physics, 47 (1990). Zbl0727.11033MR91h:11088
  14. [K] M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Math. Monographs 18 (1959). Zbl0088.10303
  15. [KS] N. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues and Monodromy, (1996), Book, to appear. Zbl0958.11004
  16. [KS] N. Katz and P. Sarnak, Zeros of zeta functions, their spacings and spectral nature, (1997), to appear. 
  17. [M] H. Montgomery, The pair correlation of zeros of the zeta function, Analytic Number Theory, AMS (1973). Zbl0268.10023MR49 #2590
  18. [O] A. Odlyzko, On the distribution of spacings between zeros of zeta functions, Math. Comp. 48 (1987), 273-308. Zbl0615.10049MR88d:11082
  19. [P] G. Pólya, Collected Papers, Cambridge, M.I.T. Press (1974). 
  20. [Pat] S. Patterson, An introduction to the theory of the Riemann zeta function, Cambridge Studies in advanced mathematics, 14 Cambridge University Press (1988). Zbl0641.10029MR89d:11072
  21. [R] B. Riemann, Mathematical Werke, Dover, New York (1953). 
  22. [W1] A. Weil, Basic Number Theory, Springer, New York (1974). Zbl0326.12001MR55 #302
  23. [W2] A. Weil, Fonctions zêta et distributions, Séminaire Bourbaki, 312, (1966). Zbl0226.12008
  24. [W3] A. Weil, Sur les formules explicites de la théorie des nombres, Izv. Mat. Nauk., (Ser. Mat.) 36, 3-18. Zbl0245.12010MR52 #345
  25. [Z] D. Zagier, Eisenstein series and the Riemannian zeta function, Automorphic Forms, Representation Theory and Arithmetic, Tata, Bombay (1979), 275-301. Zbl0484.10019MR83j:10027

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