Research PapersNo Access

RIEMANN ZETA ZEROS AND PRIME NUMBER SPECTRA IN QUANTUM FIELD THEORY

G. MENEZES

Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, SP 01140-070, Brazil

Corresponding author.

Search for more papers by this author

B. F. SVAITER

Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ 22460-320, Brazil

Search for more papers by this author

, and

N. F. SVAITER

Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, RJ 22290-180, Brazil

Search for more papers by this author

https://doi.org/10.1142/S0217751X13501285Cited by:10 (Source: Crossref)

Abstract

The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line Re(s) = 1/2. Hilbert and Pólya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of spectral theory. Using the construction of the so-called super-zeta functions or secondary zeta functions built over the Riemann nontrivial zeros and the regularity property of one of this function at the origin, we show that it is possible to extend the Hilbert–Pólya conjecture to systems with countably infinite number of degrees of freedom. The sequence of the nontrivial zeros of the Riemann zeta function can be interpreted as the spectrum of a self-adjoint operator of some hypothetical system described by the functional approach to quantum field theory. However, if one considers the same situation with numerical sequences whose asymptotic distributions are not "far away" from the asymptotic distribution of prime numbers, the associated functional integral cannot be constructed. Finally, we discuss possible relations between the asymptotic behavior of a sequence and the analytic domain of the associated zeta function.

Keywords:

PACS: 02.10.De, 11.10.-z

You currently do not have access to the full text article.
Recommend the journal to your library today!

References

B. Riemann, Über die Anzahl der Primzahlen (Monatsberichte der Berliner Akademie, 1859) p. 671. Google Scholar
P. Borwein et al. (eds.) , The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike ( Springer , New York , 2008 ) . Google Scholar
D. Schumayer and D. A. W. Hutchinson, Rev. Mod. Phys. 83, 307 (2011), DOI: 10.1103/RevModPhys.83.307. Web of Science, ADS, Google Scholar
M. G. Gutzwiller, J. Math. Phys. 11, 1791 (1970), DOI: 10.1063/1.1665328. Web of Science, ADS, Google Scholar
M. G. Gutzwiller, J. Math. Phys. 12, 343 (1971), DOI: 10.1063/1.1665596. Web of Science, ADS, Google Scholar
O. Bohigas, M. J. Giannoni and C. Schmit, Phys. Rev. Lett. 52, 1 (1984), DOI: 10.1103/PhysRevLett.52.1. Web of Science, ADS, Google Scholar
O. Bohigas and M. J. Gianonni , Mathematical and Computational Methods in Nuclear Physics , eds. H. Araki et al. ( Springer-Verlag , Berlin , 1984 ) . Google Scholar
M. V. Berry , Chaotic Behavior of Deterministic Systems , eds. G. Iooss , R. Helleman and R. Stora ( North Holland , Amsterdam , 1981 ) . Google Scholar
M. L. Mehta , Random Matrices ( Elsevier , Amsterdam , 2004 ) . Google Scholar
P. J. Forrester, N. C. Snaith and J. J. M. Verbaarschot, J. Phys. A 36, R1 (2003), DOI: 10.1088/0305-4470/36/12/201. Google Scholar
E. Brézin et al. (eds.) , Application of Random Matrices in Physics ( Springer , Netherlands , 2006 ) . Google Scholar
H. L. Montgomery, Proc. Symp. Pure Math. 24, 181 (1973), DOI: 10.1090/pspum/024/9944. Web of Science, Google Scholar
A. M. Odlyzko, Math. Comput. 48, 273 (1987), DOI: 10.1090/S0025-5718-1987-0866115-0. Web of Science, Google Scholar
E. B. Bogolmony and J. P. Keating, Nonlinearity 8, 1115 (1995). Web of Science, Google Scholar
E. B. Bogolmony and J. P. Keating, Nonlinearity 9, 911 (1996). Web of Science, Google Scholar
M. V. Berry and J. P. Keating, Siam Rev. 41, 236 (1999), DOI: 10.1137/S0036144598347497. Web of Science, ADS, Google Scholar
J. P. Keating and N. C. Snaith, Commun. Math. Phys. 214, 57 (2000), DOI: 10.1007/s002200000261. Web of Science, ADS, Google Scholar
P. Bourgade and J. P. Keating, Séminaire Poincaré 115 (2010). Google Scholar
M. V. Berry and J. P. Keating , Supersymmetry and Trace Formulae: Chaos and Disorder , eds. I. V. Lerner , J. P. Keating and D. E. Khmelnitskii ( Plenum , New York , 1999 ) . Google Scholar
M. V. Berry and J. P. Keating, Siam Rev. 41, 236 (1999), DOI: 10.1137/S0036144598347497. Web of Science, ADS, Google Scholar
G. Sierra and P. K. Townsend, Phys. Rev. Lett. 101, 110201 (2008), DOI: 10.1103/PhysRevLett.101.110201. Web of Science, Google Scholar
G. Sierra and J. Rodríguez-Laguna, Phys. Rev. Lett. 106, 200201 (2011), DOI: 10.1103/PhysRevLett.106.200201. Web of Science, Google Scholar
A. Ishimaru , Wave Propagation and Scattering in Random Media ( Academic , New York , 1978 ) . Google Scholar
P. Sheng (ed.) , Scattering and Localization of Classical Waves in Random Media ( World Scientific , Singapore , 1990 ) . Link, Google Scholar
G. Krein, G. Menezes and N. F. Svaiter, Phys. Rev. Lett. 105, 131301 (2010), DOI: 10.1103/PhysRevLett.105.131301. Web of Science, ADS, Google Scholar
E. Ariaset al., Phys. Rev. D 83, 125022 (2011), DOI: 10.1103/PhysRevD.83.125022. Web of Science, ADS, Google Scholar
L. H. Fordet al., Ann. Phys. 329, 80 (2013), DOI: 10.1016/j.aop.2012.10.002. Web of Science, ADS, Google Scholar
K. B. Efetov, Sov. Phys. JETP 55, 514 (1982). Web of Science, ADS, Google Scholar
K. B. Efetov, Sov. Phys. JETP 83, 833 (1982). Google Scholar
J. Smit , Introduction to Quantum Field Theory on a Lattice ( Cambridge University Press , Cambridge , 2002 ) . Google Scholar
C. G. Bollini and J. J. Giambiagi, Nuovo Cimento B 12, 20 (1972). ADS, Google Scholar
G. 't Hooft and M. Veltman, Nucl. Phys. B 44, 189 (1972), DOI: 10.1016/0550-3213(72)90279-9. Web of Science, Google Scholar
L. F. Ashmore, Nuovo Cimento Lett. 9, 289 (1972). Web of Science, Google Scholar
R. T. Seeley, Amer. Math. Soc. Proc. Symp. Pure Math. 10, 288 (1967), DOI: 10.1090/pspum/010/0237943. Web of Science, Google Scholar
D. B. Ray and I. M. Singer, Adv. Math. 7, 145 (1971), DOI: 10.1016/0001-8708(71)90045-4. Web of Science, Google Scholar
J. S. Dowker and R. Crithley, Phys. Rev. D 13, 3224 (1976), DOI: 10.1103/PhysRevD.13.3224. Web of Science, ADS, Google Scholar
S. W. Hawking, Commun. Math. Phys. 55, 133 (1977), DOI: 10.1007/BF01626516. Web of Science, ADS, Google Scholar
A. Voros, Commun. Math. Phys. 110, 439 (1987), DOI: 10.1007/BF01212422. Web of Science, ADS, Google Scholar
A. Voros, Adv. Stud. Pure Math. 21, 327 (1992). Google Scholar
J. R. Quine, S. H. Heydary and R. Y. Song, Trans. Amer. Math. Soc. 338, 213 (1993), DOI: 10.1090/S0002-9947-1993-1100699-1. Web of Science, Google Scholar
G. V. Dune, J. Phys. A 41, 1 (2008). Google Scholar
V. E. Landau and A. Walfisz, Rend. Circ. Mat. Palermo 44, 82 (1919), DOI: 10.1007/BF03014596. Google Scholar
C. E. Fröberg, BIT 8, 187 (1968). Google Scholar
G. Menezes and N. F. Svaiter, Quantum field theories and prime numbers spectrum, [math-ph] , arXiv:1211.5198 . Google Scholar
J. C. Andrade, Int. J. Mod. Phys. A 28, 1350072 (2013), DOI: 10.1142/S0217751X13500723. Link, Web of Science, Google Scholar
A. P. Guinand, Proc. Lond. Math. Soc. 50, 107 (1945). Web of Science, Google Scholar
J. Delsarte, J. Anal. Math. 17, 419 (1966), DOI: 10.1007/BF02788668. Web of Science, Google Scholar
I. C. Chakravarty, J. Math. Anal. Appl. 30, 280 (1970), DOI: 10.1016/0022-247X(70)90161-7. Web of Science, Google Scholar
I. C. Chakravarty, J. Math. Anal. Appl. 35, 484 (1971), DOI: 10.1016/0022-247X(71)90196-X. Web of Science, Google Scholar
A. Ivic, Bull. CXXI l'Acad. Sci. Arts 26, 39 (2003). Google Scholar
A. Voros, Ann. Inst. Fourier, Grenoble 53, 665 (2003), DOI: 10.5802/aif.1955. Web of Science, Google Scholar
P. Leboeuf, A. G. Monastra and O. Bohigas, Regul. Chaotic Dyn. 2, 205 (2001). Google Scholar
P. Leboeuf and A. G. Monastra, Nucl. Phys. A 724, 69 (2003), DOI: 10.1016/S0375-9474(03)01473-8. Web of Science, ADS, Google Scholar
B. Julia , Number Theory and Physics , eds. J.-M. Luck , D. Moussa and M. Waldschmidt ( Springer-Verlag , Berlin , 1990 ) . Google Scholar
D. Spector, Commun. Math. Phys. 127, 239 (1990), DOI: 10.1007/BF02096755. Web of Science, ADS, Google Scholar
I. Bakas and M. J. Bowick, J. Math. Phys. 32, 1881 (1991), DOI: 10.1063/1.529511. Web of Science, Google Scholar
B. L. Julia, Physica A 203, 425 (1994), DOI: 10.1016/0378-4371(94)90008-6. Web of Science, ADS, Google Scholar
D. Spector, Commun. Math. Phys. 177, 13 (1996), DOI: 10.1007/BF02102428. Web of Science, ADS, Google Scholar
D. Spector, J. Math. Phys. 39, 1919 (1998), DOI: 10.1063/1.532269. Web of Science, Google Scholar
J. Hadamard, Bull. Soc. Math. France 24, 119 (1986). Google Scholar
C. de la Vallée-Poussin, Ann. Soc. Sci. Bruxelles 202, 183 (1896). Google Scholar
C. de la Vallée-Poussin, Ann. Soc. Sci. Bruxelles 202, 281 (1896). Google Scholar
A. E. Ingham , The Distribution of Prime Numbers ( Cambridge University Press , Cambridge , 1990 ) . Google Scholar
E. C. Titchmarsh , The Zeta-Function of Riemann ( Stechert-Hafner Service Agency , New York , 1964 ) . Google Scholar
S. J. Patterson , An Introduction to the Theory of the Riemann Zeta-Function ( Cambridge University Press , Cambridge , 1988 ) . Google Scholar
G. Tenenbaum and M. M. France , The Prime Numbers and Their Distribution ( American Mathematical Society , 2000 ) . Google Scholar
G. Mussardo, The quantum mechanical potential for the prime numbers, Preprint ISAS/EP/97/153, IC/97/199 , arXiv:cond-mat/9712010 . Google Scholar
H. Rosu, Mod. Phys. Lett. 18, 1205 (2003), DOI: 10.1142/S0217732303011034. Link, Web of Science, ADS, Google Scholar
J. Sakhr, R. K. Bhaduri and B. P. van Zyl, Phys. Rev. E 68, 026206 (2003), DOI: 10.1103/PhysRevE.68.026206. Web of Science, Google Scholar
D. Schumayer, B. P. van Zyl and D. A. W. Hutchinson, Phys. Rev. E 78, 056215 (2008), DOI: 10.1103/PhysRevE.78.056215. Web of Science, Google Scholar
G. H. Hardy and E. M. Wright , An Introduction to the Theory of Numbers ( Clarendon Press , Oxford, London , 1956 ) . Google Scholar
S. Minakshisundaram and M. Pleijel, Can. J. Math. 1, 242 (1949), DOI: 10.4153/CJM-1949-021-5. Web of Science, Google Scholar
A. Voros , Zeta Functions over Zeros of Zeta Functions ( Springer-Verlag , Heidelberg , 2010 ) . Google Scholar
R. Courant and D. Hilbert , Methods of Mathematical Physics ( Interscience Publishers , New York , 1953 ) . Google Scholar
D. Deutsch and P. Candelas, Phys. Rev. D 20, 3063 (1979), DOI: 10.1103/PhysRevD.20.3063. Web of Science, ADS, Google Scholar