Advanced search
Publication Cover
Experimental Mathematics Volume 15, 2006 - Issue 4
Submit an article Journal homepage
80
Views
23
CrossRef citations to date
0
Altmetric
Original Article

Artin's Conjecture, Turing's Method, and the Riemann Hypothesis

Andrew R. Booker Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom
Pages 385-407 | Published online: 30 Jan 2011

  • J. V. Armitage. "Zeta Functions with a Zero at s = ½." Invent. Math. 16 (1972), 195–205.
  • E. Artin. "Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren." Abh. Math. Sem. Univ. Hamburg 8 (1930), 292–306.
  • Andrew R. Booker. "Poles of Artin L-Functions and the Strong Artin Conjecture." Ann. of Math. (2) 158:3 (2003), 1089–1098.
  • Andrew R. Booker. "Numerical Tests of Modularity." J. Ramanujan Math. Soc. 20:4 (2005), 283–339.
  • A. R. Booker and A. Strömbergsson. "Numerical Computations with the Trace Formula and the Selberg Eigenvalue Conjecture." To appear in J. Reine Angew. Math., 2007.
  • A. R. Booker, A. Strömbergsson, and A. Venkatesh. "Effective Computation of Maass Cusp Forms." Int. Math. Res. Not. 2006 (2006), Article ID 71281, 34 pages.
  • R. Brauer. "On Artin's L-Series with General Group Characters." Ann. Math. 48 (1947), 502–514.
  • J. P. Buhler. Icosahedral Galois Representations, Lecture Notes in Mathematics, 654. Berlin: Springer- Verlag, 1978.
  • K. Buzzard and W. Stein. "A Mod Five Approach to Modularity of Icosahedral Galois Representations." Pacific Jour. Math. 203:2 (2002), 265–282.
  • K. Buzzard, M. Dickinson, N. Shepard- Barron, and R. Taylor. "On Icosahedral Artin Representations." Duke Math. J. 109 (2001), 283–318.
  • Pierre Deligne and Jean-Pierre Serre. "Formes modulaires de poids 1." Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530.
  • Tim Dokchitser. "Computing Special Values of Motivic L-Functions. Experiment. Math. 13:2 (2004), 137–149.
  • D. S. Dummit. "Solving Solvable Quintics." Math. Comp. 57:195 (1991), 387–401.
  • Yuval Z. Flicker. "On the Symmetric Square: Total Global Comparison." J. Funct. Anal. 122:2 (1994), 255–278.
  • Richard Foote and V. Kumar Murty. "Zeros and Poles of Artin L-Series." Math. Proc. Cambridge Philos. Soc. 105:1 (1989), 5–11.
  • The GAP Group. "GAP: Groups, Algorithms, and Programming, Version 4.4." Available online (http: //www.gap-system.org), 2005. http://www.gap-system.org
  • H. Iwaniec and P. Sarnak. "Perspectives on the Analytic Theory of L-Functions." Geom. Funct. Anal. (Special Volume, Part II) (2000), 705–741.
  • A. Jehanne and M. Müller. "Modularity of an Odd Icosahedral Representation." J. Théor. Nombres Bordeaux 12:2 (2000), 475–482.
  • A. Jehanne and M. Müller. "Modularity of Some Odd Icosahedral Representations." Manuscript available online (http: //www.math.u-bordeaux1.fr/~jehanne/travaux.html), 2001. http://www.math.u-bordeaux1.fr/~jehanne/travaux.html
  • I. Kiming. "On the Experimental Verification of the Artin Conjecture for 2-Dimensional Odd Galois Representations over Q. Liftings of 2-Dimensional Projective Galois Representations over Q." In On Artin's Conjecture for Odd 2-Dimensional Representations, 1–36, Lecture Notes in Math. 1585. Berlin: Springer-Verlag, 1994.
  • Jürgen Klüners and Gunter Malle. "A Database for Number Fields." Available online (http://www.mathematik.uni-kassel.de/~klueners/minimum/minimum.html), 2001. http://www.mathematik.uni-kassel.de/~klueners/minimum/minimum.html
  • J. C. Lagarias and A. M. Odlyzko. "On Computing Artin L-Functions in the Critical Strip." Math. Comp. 33:147 (1979), 1081–1095.
  • R.P. Langlands. Base Change for GL(2). Princeton: Princeton University Press, 1980.
  • R. Sherman Lehman. "On the Distribution of Zeros of the Riemann Zeta-Function." Proc. London Math. Soc. (3) 20 (1970), 303–320.
  • Wenzhi Luo, Zeév Rudnick, and Peter Sarnak. "On the Generalized Ramanujan Conjecture for GL(n)." In Automorphic Forms, Automorphic Representations, and Arithmetic, pp. 301–310, Proc. Sympos. Pure Math. 66. Providence: Amer. Math. Soc., 1999.
  • A. M. Odlyzko. "On the Distribution of Spacings between Zeros of the Zeta Function." Math. Comp. 48 (1987), 273–308.
  • A. M. Odlyzko and A. Schönhage. "Fast Algorithms for Multiple Evaluations of the Riemann Zeta Function." Trans. Amer. Math. Soc. 309:2 (1988), 797–809.
  • Sami Omar. "Localization of the First Zero of the Dedekind Zeta Function." Math. Comp. 70:236 (2001), 1607–1616.
  • The PARI Group, Bordeaux. "PARI/- GP, version 2.1.5." Available online (http://pari.math. u-bordeaux.fr/), 2004. http://pari.math.u-bordeaux.fr/
  • Nathalie Revol and Fabrice Rouiller. "Multiple Precision Floating-Point Interval Library, Version 1.3.3." Available online (http://perso.ens-lyon.fr/nathalie.revol/software.html), 2005. http://perso.ens-lyon.fr/nathalie.revol/software.html
  • Michael Rubinstein. "Computational Methods and Experiments in Analytic Number Theory." In Recent Perspectives in Random Matrix Theory and Number Theory, pp. 425–506, London Math. Soc. Lecture Note Ser. 322. Cambridge: Cambridge Univ. Press, 2005.
  • Zeév Rudnick and Peter Sarnak. "Zeros of Principal L-Functions and Random Matrix Theory." Duke Math. J. 81:2 (1996), 269–322.
  • Robert Rumely. "Numerical Computations Concerning the ERH." Math. Comp. 61:203 (1993), 415–440, S17–S23.
  • Richard Taylor. "On Icosahedral Artin Representations II." Amer. J. Math. 125:3 (2003), 549–566.
  • E. C. Titchmarsh. The Theory of the Riemann Zeta-Function, second edition. New York: Oxford University Press, 1986.
  • Emmanuel Tollis. "Zeros of Dedekind Zeta Functions in the Critical Strip." Math. Comp. 66:219 (1997), 1295–1321.
  • J. Tunnell. "Artin's Conjecture for Representations of Octahedral Type." Bull. AMS 5 (1981), 173–175.
  • A. M. Turing. "Some Calculations of the Riemann Zeta-Function." Proc. London Math. Soc. (3) 3 (1953), 99–117.
  • Song Wang. "On the Symmetric Powers of Cusp Forms on GL(2) of Icosahedral Type." Int. Math. Res. Not. 44 (2003), 2373–2390.
  • M. Weissman. "Icosahedral Galois Representations and Modular Forms." Bachelor's thesis, Princeton University, 1999.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Order Reprints Request Corporate Permissions

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

Request Academic Permissions

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.