On recent strategies proposed for proving the Riemann hypothesis

International Journal of Modern Physics A

Volumn 18, Issue 12, 2003, Pages 2189-2195

  Elizalde, E ^a,c   Moretti, V ^b   Zerbini, S ^b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF TRENTO   (Italy)

^c INSTITUT D ESTUDIS ESPACIALS DE CATALUNYA IEEC   (Spain)

Author keywords

Orthogonality; Riemann hypothesis; Zeta function regularization

Indexed keywords

CONFERENCE PAPER; MATHEMATICAL ANALYSIS; OPERATOR; PHYSICAL PARAMETERS; PHYSICS; QUANTUM THEORY;

EID: 0038475895     PISSN: 0217751X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0217751X03015647     Document Type: Conference Paper

References (37)

1. B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monat. der Königl. Preuss. Akad. der Wissen. zu Berlin aus der Jahre 1859, 671 (1860); Gesammelte Math. Werke und wissensch. Nachlass. 2, 145 (1892). (See also the documented web page by M. Watkins, www.maths.ex.ac.uk/~mwatkins/zeta/).
2. B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monat. der Königl. Preuss. Akad. der Wissen. zu Berlin aus der Jahre 1859, 671 (1860); Gesammelte Math. Werke und wissensch. Nachlass. 2, 145 (1892). (See also the documented web page by M. Watkins, www.maths.ex.ac.uk/~mwatkins/zeta/).
3. B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monat. der Königl. Preuss. Akad. der Wissen. zu Berlin aus der Jahre 1859, 671 (1860); Gesammelte Math. Werke und wissensch. Nachlass. 2, 145 (1892). (See also the documented web page by M. Watkins, www.maths.ex.ac.uk/~mwatkins/zeta/).
4. M. Pitkänen, Riemann hypothesis and superconformal invariance, math.GM/0102031.
5. C. Castro, On p-adic stochastic dynamics, supersymmetry and the Riemann conjecture, physics/0101104.
6. C. Castro, A. Granik, and J. Mahecha, On SUSY-QM, fractal strings and steps towards a proof of the Riemann hypothesis, hep-th/0107266.
7. E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994); E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer-Verlag, Berlin, 1995); A. A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini, Phys. Reports 266, 1 (1996); E. Elizalde, Commun. Math. Phys. 198, 83 (1998); V. Moretti, Commun. Math. Phys. 201, 327 (1999); V. Moretti, J. Math. Phys. 40, 3843 (1999); E. Elizalde, J. Comput. Appl. Math. 118, 125 (2000).
8. E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994); E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer-Verlag, Berlin, 1995); A. A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini, Phys. Reports 266, 1 (1996); E. Elizalde, Commun. Math. Phys. 198, 83 (1998); V. Moretti, Commun. Math. Phys. 201, 327 (1999); V. Moretti, J. Math. Phys. 40, 3843 (1999); E. Elizalde, J. Comput. Appl. Math. 118, 125 (2000).
9. E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994); E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer-Verlag, Berlin, 1995); A. A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini, Phys. Reports 266, 1 (1996); E. Elizalde, Commun. Math. Phys. 198, 83 (1998); V. Moretti, Commun. Math. Phys. 201, 327 (1999); V. Moretti, J. Math. Phys. 40, 3843 (1999); E. Elizalde, J. Comput. Appl. Math. 118, 125 (2000).
10. E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994); E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer-Verlag, Berlin, 1995); A. A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini, Phys. Reports 266, 1 (1996); E. Elizalde, Commun. Math. Phys. 198, 83 (1998); V. Moretti, Commun. Math. Phys. 201, 327 (1999); V. Moretti, J. Math. Phys. 40, 3843 (1999); E. Elizalde, J. Comput. Appl. Math. 118, 125 (2000).
11. E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994); E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer-Verlag, Berlin, 1995); A. A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini, Phys. Reports 266, 1 (1996); E. Elizalde, Commun. Math. Phys. 198, 83 (1998); V. Moretti, Commun. Math. Phys. 201, 327 (1999); V. Moretti, J. Math. Phys. 40, 3843 (1999); E. Elizalde, J. Comput. Appl. Math. 118, 125 (2000).
12. E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994); E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer-Verlag, Berlin, 1995); A. A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini, Phys. Reports 266, 1 (1996); E. Elizalde, Commun. Math. Phys. 198, 83 (1998); V. Moretti, Commun. Math. Phys. 201, 327 (1999); V. Moretti, J. Math. Phys. 40, 3843 (1999); E. Elizalde, J. Comput. Appl. Math. 118, 125 (2000).
13. E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994); E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer-Verlag, Berlin, 1995); A. A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini, Phys. Reports 266, 1 (1996); E. Elizalde, Commun. Math. Phys. 198, 83 (1998); V. Moretti, Commun. Math. Phys. 201, 327 (1999); V. Moretti, J. Math. Phys. 40, 3843 (1999); E. Elizalde, J. Comput. Appl. Math. 118, 125 (2000).
14. J. Main, Phys. Rep. 316, 233 (1999); H. Montgomery, Proc. Int. Congress of Mathematics, Vancouver, 1, 379 (1974); M. Berry and J. Keating, SIAM Review 41, 236 (1999); A. Connes, Selecta Math. (NS) 5, 29 (1999).
15. J. Main, Phys. Rep. 316, 233 (1999); H. Montgomery, Proc. Int. Congress of Mathematics, Vancouver, 1, 379 (1974); M. Berry and J. Keating, SIAM Review 41, 236 (1999); A. Connes, Selecta Math. (NS) 5, 29 (1999).
16. J. Main, Phys. Rep. 316, 233 (1999); H. Montgomery, Proc. Int. Congress of Mathematics, Vancouver, 1, 379 (1974); M. Berry and J. Keating, SIAM Review 41, 236 (1999); A. Connes, Selecta Math. (NS) 5, 29 (1999).
17. J. Main, Phys. Rep. 316, 233 (1999); H. Montgomery, Proc. Int. Congress of Mathematics, Vancouver, 1, 379 (1974); M. Berry and J. Keating, SIAM Review 41, 236 (1999); A. Connes, Selecta Math. (NS) 5, 29 (1999).
18. L. de Branges, J. Funct. Anal. 107, 122 (1992); J. Alcántara-Bode, Int. Eqs. Op. Theory 17, 151 (1993); L. de Branges, J. Funct. Anal. 121, 117 (1994); X.-J. Li, Math. Nachr. 166, 229 (1994); N. Nikolski, Ann. Inst. Fourier (Grenoble) 45, 143 (1995); V.I. Vasyunin, Algebra i Analiz 7, 118 (1995) [trans. St. Petersburg Math. J. 7, 405 (1996)].
19. L. de Branges, J. Funct. Anal. 107, 122 (1992); J. Alcántara-Bode, Int. Eqs. Op. Theory 17, 151 (1993); L. de Branges, J. Funct. Anal. 121, 117 (1994); X.-J. Li, Math. Nachr. 166, 229 (1994); N. Nikolski, Ann. Inst. Fourier (Grenoble) 45, 143 (1995); V.I. Vasyunin, Algebra i Analiz 7, 118 (1995) [trans. St. Petersburg Math. J. 7, 405 (1996)].
20. L. de Branges, J. Funct. Anal. 107, 122 (1992); J. Alcántara-Bode, Int. Eqs. Op. Theory 17, 151 (1993); L. de Branges, J. Funct. Anal. 121, 117 (1994); X.-J. Li, Math. Nachr. 166, 229 (1994); N. Nikolski, Ann. Inst. Fourier (Grenoble) 45, 143 (1995); V.I. Vasyunin, Algebra i Analiz 7, 118 (1995) [trans. St. Petersburg Math. J. 7, 405 (1996)].
21. L. de Branges, J. Funct. Anal. 107, 122 (1992); J. Alcántara-Bode, Int. Eqs. Op. Theory 17, 151 (1993); L. de Branges, J. Funct. Anal. 121, 117 (1994); X.-J. Li, Math. Nachr. 166, 229 (1994); N. Nikolski, Ann. Inst. Fourier (Grenoble) 45, 143 (1995); V.I. Vasyunin, Algebra i Analiz 7, 118 (1995) [trans. St. Petersburg Math. J. 7, 405 (1996)].
22. L. de Branges, J. Funct. Anal. 107, 122 (1992); J. Alcántara-Bode, Int. Eqs. Op. Theory 17, 151 (1993); L. de Branges, J. Funct. Anal. 121, 117 (1994); X.-J. Li, Math. Nachr. 166, 229 (1994); N. Nikolski, Ann. Inst. Fourier (Grenoble) 45, 143 (1995); V.I. Vasyunin, Algebra i Analiz 7, 118 (1995) [trans. St. Petersburg Math. J. 7, 405 (1996)].
23. L. de Branges, J. Funct. Anal. 107, 122 (1992); J. Alcántara-Bode, Int. Eqs. Op. Theory 17, 151 (1993); L. de Branges, J. Funct. Anal. 121, 117 (1994); X.-J. Li, Math. Nachr. 166, 229 (1994); N. Nikolski, Ann. Inst. Fourier (Grenoble) 45, 143 (1995); V.I. Vasyunin, Algebra i Analiz 7, 118 (1995) [trans. St. Petersburg Math. J. 7, 405 (1996)].
24. L. de Branges, J. Funct. Anal. 107, 122 (1992); J. Alcántara-Bode, Int. Eqs. Op. Theory 17, 151 (1993); L. de Branges, J. Funct. Anal. 121, 117 (1994); X.-J. Li, Math. Nachr. 166, 229 (1994); N. Nikolski, Ann. Inst. Fourier (Grenoble) 45, 143 (1995); V.I. Vasyunin, Algebra i Analiz 7, 118 (1995) [trans. St. Petersburg Math. J. 7, 405 (1996)].
25. E. Elizalde, J. Phys. A27, L299 (1994); 3755 (1994).
26. E. Elizalde, J. Phys. A27, L299 (1994); 3755 (1994).
27. E. Elizalde, JHEP 9907, 015 (1999).
28. E. Elizalde, L. Vanzo and S. Zerbini, Commun. Math. Phys. 194, 613 (1998); E. Elizalde, G. Cognola and S. Zerbini, Nucl. Phys. B 532, 407 (1998); J. J. McKenzie-Smith and D. J. Toms, Phys. Rev. D 58, 105001 (1998); A. A. Bytsenko and F. L. Williams, J. Math. Phys. 39, 1075 (1998); N. Evans, Phys. Lett. B 457, 127 (1999); J. S. Dowker, On the relevance on the multiplicative anomaly, hep-th/9803200; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly dependent on the regularization?, hep-th/9804071; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly relevant?, hep-th/9804072.
29. E. Elizalde, L. Vanzo and S. Zerbini, Commun. Math. Phys. 194, 613 (1998); E. Elizalde, G. Cognola and S. Zerbini, Nucl. Phys. B 532, 407 (1998); J. J. McKenzie-Smith and D. J. Toms, Phys. Rev. D 58, 105001 (1998); A. A. Bytsenko and F. L. Williams, J. Math. Phys. 39, 1075 (1998); N. Evans, Phys. Lett. B 457, 127 (1999); J. S. Dowker, On the relevance on the multiplicative anomaly, hep-th/9803200; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly dependent on the regularization?, hep-th/9804071; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly relevant?, hep-th/9804072.
30. E. Elizalde, L. Vanzo and S. Zerbini, Commun. Math. Phys. 194, 613 (1998); E. Elizalde, G. Cognola and S. Zerbini, Nucl. Phys. B 532, 407 (1998); J. J. McKenzie-Smith and D. J. Toms, Phys. Rev. D 58, 105001 (1998); A. A. Bytsenko and F. L. Williams, J. Math. Phys. 39, 1075 (1998); N. Evans, Phys. Lett. B 457, 127 (1999); J. S. Dowker, On the relevance on the multiplicative anomaly, hep-th/9803200; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly dependent on the regularization?, hep-th/9804071; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly relevant?, hep-th/9804072.
31. E. Elizalde, L. Vanzo and S. Zerbini, Commun. Math. Phys. 194, 613 (1998); E. Elizalde, G. Cognola and S. Zerbini, Nucl. Phys. B 532, 407 (1998); J. J. McKenzie-Smith and D. J. Toms, Phys. Rev. D 58, 105001 (1998); A. A. Bytsenko and F. L. Williams, J. Math. Phys. 39, 1075 (1998); N. Evans, Phys. Lett. B 457, 127 (1999); J. S. Dowker, On the relevance on the multiplicative anomaly, hep-th/9803200; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly dependent on the regularization?, hep-th/9804071; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly relevant?, hep-th/9804072.
32. E. Elizalde, L. Vanzo and S. Zerbini, Commun. Math. Phys. 194, 613 (1998); E. Elizalde, G. Cognola and S. Zerbini, Nucl. Phys. B 532, 407 (1998); J. J. McKenzie-Smith and D. J. Toms, Phys. Rev. D 58, 105001 (1998); A. A. Bytsenko and F. L. Williams, J. Math. Phys. 39, 1075 (1998); N. Evans, Phys. Lett. B 457, 127 (1999); J. S. Dowker, On the relevance on the multiplicative anomaly, hep-th/9803200; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly dependent on the regularization?, hep-th/9804071; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly relevant?, hep-th/9804072.
33. E. Elizalde, L. Vanzo and S. Zerbini, Commun. Math. Phys. 194, 613 (1998); E. Elizalde, G. Cognola and S. Zerbini, Nucl. Phys. B 532, 407 (1998); J. J. McKenzie-Smith and D. J. Toms, Phys. Rev. D 58, 105001 (1998); A. A. Bytsenko and F. L. Williams, J. Math. Phys. 39, 1075 (1998); N. Evans, Phys. Lett. B 457, 127 (1999); J. S. Dowker, On the relevance on the multiplicative anomaly, hep-th/9803200; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly dependent on the regularization?, hep-th/9804071; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly relevant?, hep-th/9804072.
34. E. Elizalde, L. Vanzo and S. Zerbini, Commun. Math. Phys. 194, 613 (1998); E. Elizalde, G. Cognola and S. Zerbini, Nucl. Phys. B 532, 407 (1998); J. J. McKenzie-Smith and D. J. Toms, Phys. Rev. D 58, 105001 (1998); A. A. Bytsenko and F. L. Williams, J. Math. Phys. 39, 1075 (1998); N. Evans, Phys. Lett. B 457, 127 (1999); J. S. Dowker, On the relevance on the multiplicative anomaly, hep-th/9803200; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly dependent on the regularization?, hep-th/9804071; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly relevant?, hep-th/9804072.
35. E. Elizalde, L. Vanzo and S. Zerbini, Commun. Math. Phys. 194, 613 (1998); E. Elizalde, G. Cognola and S. Zerbini, Nucl. Phys. B 532, 407 (1998); J. J. McKenzie-Smith and D. J. Toms, Phys. Rev. D 58, 105001 (1998); A. A. Bytsenko and F. L. Williams, J. Math. Phys. 39, 1075 (1998); N. Evans, Phys. Lett. B 457, 127 (1999); J. S. Dowker, On the relevance on the multiplicative anomaly, hep-th/9803200; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly dependent on the regularization?, hep-th/9804071; E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini, Is the multiplicative anomaly relevant?, hep-th/9804072.
36. M. Wodzicki. Noncommutative Residue, Chapter I, in Lecture Notes in Mathematics, ed. Yu. I. Manin 1289, 320 (Springer-Verlag, Berlin, 1987); C. Kassel, Asterisque 177, 199 (1989), Sem. Bourbaki.
37. M. Wodzicki. Noncommutative Residue, Chapter I, in Lecture Notes in Mathematics, ed. Yu. I. Manin 1289, 320 (Springer-Verlag, Berlin, 1987); C. Kassel, Asterisque 177, 199 (1989), Sem. Bourbaki.