Research ArticlesNo Access

THE RIEMANN ZETA FUNCTION AS AN EQUIVARIANT DIRAC INDEX

MAURO SPERA

Dipartimento di Informatica, Università degli Studi di Verona, Ca' Vignal 2, Strada le Grazie 15, I-37134 Verona, Italy

https://doi.org/10.1142/S0219887812500715Cited by:1 (Source: Crossref)

Abstract

In this note an interpretation of Riemann's zeta function is provided in terms of an ℝ-equivariant L²-index of a Dirac–Ramond type operator, akin to the one on (mean zero) loops in flat space constructed by the present author and T. Wurzbacher. We build on the formal similarity between Euler's partitio numerorum function (the S¹-equivariant L²-index of the loop space Dirac–Ramond operator) and Riemann's zeta function. Also, a Lefschetz–Atiyah–Bott interpretation of the result together with a generalization to M. Lapidus' fractal membranes are also discussed. A fermionic Bost–Connes type statistical mechanical model is presented as well, exhibiting a "phase transition at (inverse) temperature β = 1", which also holds for some "well-behaved" g-prime systems in the sense of Hilberdink–Lapidus.

Keywords:

AMSC: 47B25, 19K56, 11M06, 46L87, 46L36, 11N80

References

O. Alvarez, Geometry and Quantum Field Theory, eds. D. Freed and K. Uhlenbeck (American Mathematical Society, Providence, RI, 1995) pp. 271–322. Crossref, Google Scholar
O. Alvarez and P. Windey, Proc. Natl. Acad. Sci. USA 107(11), 4845 (2010), DOI: 10.1073/pnas.0915124107. Crossref, Web of Science, Google Scholar
H. Araki and E. J. Woods, Publ. Res. Inst. Math. Sci. Kyoto Univ. Ser. A 3, 51 (1968). Google Scholar
M. F. Atiyah and R. Bott, Ann. Math. 86, 374 (1967), DOI: 10.2307/1970694. Crossref, Web of Science, Google Scholar
N. Berline , E. Getzler and M. Vergne , Heat Kernels and Dirac Operators ( Springer , Berlin , 2004 ) . Google Scholar
F. Boca and A. Zaharescu, Proc. London Math. Soc. (3) 80(1), 145 (2000), DOI: 10.1112/S0024611500012193. Crossref, Web of Science, Google Scholar
J.-B. Bost and A. Connes, C. R. Math. Acad. Sci. Paris Ser. I 315, 279 (1992). Web of Science, Google Scholar
J.-B. Bost and A. Connes, Selecta Math. (NS) 1, 411 (1995), DOI: 10.1007/BF01589495. Crossref, Google Scholar
A. Connes, Ann. Sci. École Norm. Sup. (4) 6(2), 133 (1973). Crossref, Google Scholar
A. Connes, Sur la théorie non commutative de l'intégration, in Algèbres d'opérateurs, Lecture Notes in Mathematics, Vol. 725 (Springer, Berlin, 1979), pp. 19–143 . Google Scholar
A. Connes , Noncommutative Geometry ( Academic Press , San Diego , 1994 ) . Google Scholar
A. Connes and M. Marcolli , Noncommutative Geometry, Quantum Fields and Motives ( American Mathematical Society , Providence, RI , 2008 ) . Google Scholar
H. W. Edwards , Riemann's Zeta Function ( Dover , New York , 2001 ) . Google Scholar
E. Getzler, Topology 25, 111 (1986), DOI: 10.1016/0040-9383(86)90008-X. Crossref, Web of Science, Google Scholar
A. Guichardet, Ann. Sci. École Norm. Sup. (3) 83, 1 (1966). Crossref, Google Scholar
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edn. (Oxford University Press, Oxford, 1979). Google Scholar
T. W. Hilberdink and M. L. Lapidus, Acta Appl. Math. 94, 21 (2006), DOI: 10.1007/s10440-006-9063-0. Crossref, Web of Science, Google Scholar
K. Kraus, L. Polley and G. Reents, J. Math. Phys. 18(11), 2166 (1977), DOI: 10.1063/1.523183. Crossref, Web of Science, Google Scholar
M. Lapidus , In Search of the Riemann Zeros ( American Mathematical Society , Providence, RI , 2008 ) . Crossref, Google Scholar
R. Powers and E. Størmer, Commun. Math. Phys. 16, 1 (1970), DOI: 10.1007/BF01645492. Crossref, Web of Science, Google Scholar
M. Reed and B. Simon , Methods of Modern Mathematical Physics, Vol. I: Functional Analysis ( Academic Press , New York , 1972 ) . Google Scholar
M. Reed and B. Simon , Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis, Self-Adjointness ( Acadamic Press , New York , 1975 ) . Google Scholar
M. Reed and B. Simon , Methods of Modern Mathematical Physics, Vol. 3: Scattering Theory ( Academic Press , New York , 1979 ) . Google Scholar
J. Roe , Elliptic Operators, Topology and Asymptotic Methods , 2nd edn. ( Longman, Harlow , 1998 ) . Google Scholar
M. Spera and T. Wurzbacher, J. Funct. Anal. 197, 110 (2003), DOI: 10.1016/S0022-1236(02)00178-7. Crossref, Web of Science, Google Scholar
L. Streit, Commun. Math. Phys. 4, 22 (1967), DOI: 10.1007/BF01645175. Crossref, Google Scholar
M. Takesaki , Theory of Operator Algebras, Part I ( Springer , New York , 2002 ) . Google Scholar
M. Takesaki , Theory of Operator Algebra, Part II ( Springer , New York , 2003 ) . Crossref, Google Scholar
M. Takesaki , Theory of Operator Algebra, Part III ( Springer , New York , 2003 ) . Crossref, Google Scholar
K. Tzanev, Hecke pairs and type III factors, preprint (2004) (unpublished: available online) . Google Scholar
J. von Neumann, On infinite direct products, Compos. Math. 6 (1939) 1–77. Also in: Collected Works III (Pergamon Press, New York, 1961), pp. 323–399 . Google Scholar