Models of Computation, Riemann Hypothesis, and Classical Mathematics

  • RŪsiņš Freivalds
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1521)

Abstract

Classical mathematics is a source of ideas used by Computer Science since the very first days. Surprisingly, there is still much to be found. Computer scientists, especially, those in Theoretical Computer Science find inspiring ideas both in old notions and results, and in the 20th century mathematics. The latest decades have brought us evidence that computer people will soon study quantum physics and modern biology just to understand what computers are doing.

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References

  1. 1.
    Leonard Adleman. Molecular computation of solutions to combinatorial problems. Science, 1994, vol. 266, p. 1021–1024.CrossRefGoogle Scholar
  2. 2.
    Andris Ambainis and RŪsiņš Freivalds. 1-way quantum finite automata: strengths, weaknesses and generalizations. Proc. 39th FOCS, 1998 http://xxx.lanl.gov/abs/quant-ph/9802062
  3. 3.
    N. C. Ankeny. The least quadratic non residue. Annals of Mathematics, 1952, vol. 55, p. 65–72.CrossRefMathSciNetGoogle Scholar
  4. 4.
    Eric Bach. Fast algorithms under the Extended Riemann Hypothesis: a concrete estimate. Proc. 14th STOC, 1982, p. 290–295.Google Scholar
  5. 5.
    Eric Bach. Realistic analysis of some randomized algorithms. Journal of Computer and System Sciences, 1991, vol. 42, No. 1, p. 30–53.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Paul Benioff. Quantum mechanical Hamiltonian models of Turing machines. J. Statistical Physics, 1982, vol. 29, p. 515–546.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Michael Ben-Or. Probabilistic algorithms in finite fields. Proc. 22nd FOCS, 1981, p. 394–398.Google Scholar
  8. 8.
    Johannes Buchmann and Victor Shoup. Constructing nonresidues in finite fields and the Extended Riemann Hypothesis. Proc. 23rd STOC, 1991, p. 72–79.Google Scholar
  9. 9.
    Alan Cobham. The Recognition Problem for the Set of Perfect Squares. Proc. FOCS, 1966, p. 78–87.Google Scholar
  10. 10.
    Pierre Deligne. La conjeture de Weil. Publ. Math. Inst. HES, v. 43, 1974, p. 273–307.CrossRefMathSciNetGoogle Scholar
  11. 11.
    David Deutsch. Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. Royal Society London, A400, 1989. p. 96–117.Google Scholar
  12. 12.
    Richard Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 1982, vol. 21, No. 6/7, p. 467–488.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Ernests Fogels. On the zeros of L-functions. Acta Arithmetica, 1965, vol. 11, p. 67–96.MATHMathSciNetGoogle Scholar
  14. 14.
    RŪsiņš Freivalds. Fast computations by probabilistic Turing machines. In Theory of Algorithms and Programs, J. Bārzdiņš, Ed., University of Latvia, Riga, 1975, p. 3–34 (in Russian).Google Scholar
  15. 15.
    RŪsiņš Freivalds. Recognition of languages with high probability by various types of automata. Dokladi AN SSSR, 1978, vol. 239, No. 1, p. 60–62 (in Russian).MathSciNetGoogle Scholar
  16. 16.
    RŪsiņš Freivalds. Fast probabilistic algorithms. Lecture Notes in Computer Science, 1979, vol. 74, p. 57–69.Google Scholar
  17. 17.
    RŪsiņš Freivalds. Probabilistic two-way machines. Lecture Notes in Computer Science, 1981, vol. 118, p. 33–45.Google Scholar
  18. 18.
    RŪsiņš Freivalds. Space and reversal complexity of probabilistic one-way Turing machines. Annals of Discrete Mathematics, 1985, vol. 24, p. 39–50.MathSciNetGoogle Scholar
  19. 19.
    RŪsiņš Freivalds and Marek Karpinski. Lower space bounds for randomized computation. Lecture Notes in Computer Science, 1994, vol. 820, p. 580–592.Google Scholar
  20. 20.
    RŪsiņš Freivalds and Marek Karpinski. Lower time bounds for randomized computation. Lecture Notes in Computer Science, 1995, vol. 944, p. 154–168.Google Scholar
  21. 21.
    Dima Grigoriev, Marek Karpinski and Andrew M. Odlyzko. Existence of short proofs for nondivisibility of sparse polynomials under the Extended Riemann Hypothesis. Proc. Int. Symp. on Symbolic and Algebraic Computation, 1992, p. 117–122.Google Scholar
  22. 22.
    Ming-Deh A. Huang. Riemann Hypothesis and finding roots over finite fields. Proc. 17th STOC, 1985, p. 121–130.Google Scholar
  23. 23.
    Ming-Deh A. Huang. Generalized Riemann Hypothesis and factoring polynomials over finite fields. Journal of Algorithms, 1991,vol. 12, No. 3, p. 464–481.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kenneth Ireland and Michael Rosen. A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Heidelberg-Berlin, 1972.Google Scholar
  25. 25.
    Lila Kari. DNA computers, tomorrow’s reality. Bulletin of the EATCS, vol. 59, p. 256–266.Google Scholar
  26. 26.
    Marek Karpinski and Rutger Verbeek. On randomized versus deterministic computation. Lecture Notes in Computer Science, 1993, vol. 700, p. 227–240.Google Scholar
  27. 27.
    Attila Kondacs and John Watrous. On the power of quantum finite state automata. In Proc. 38th FOCS, 1997, p. 66–75.Google Scholar
  28. 28.
    K. de Leeuw, E.F. Moore, C.E. Shannon and N. Shapiro. Computability by probabilistic machines. In Automata Studies, C.E. Shannon and J. McCarthy, Eds., Princeton University Press, Princeton, NJ, 1955, p. 183–212.Google Scholar
  29. 29.
    R.F. Lukes, C.D. Paterson, and H.C. Williams. Some Results on Pseudosquares. Mathematics of Computation, 1996, v. 65, No. 213, p. 361–372.MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Gary L. Miller. Riemann’s hypothesis and tests for primality. Journal of Computer and System Sciences, 1976, vol. 13, No. 3, p. 300–317.MATHMathSciNetGoogle Scholar
  31. 31.
    Hugh L. Montgomery. Topics in Multiplicative Number Theory. Lecture Notes in Mathematics, 1971. vol. 227.Google Scholar
  32. 32.
    Cristopher Moore, James P. Crutchfield Quantum automata and quantum grammars. Manuscript available at http://xxx.lanl.gov/abs/quant-ph/9707031
  33. 33.
    Max Planck. Uber eine Verbesserung der Wien’schen Spectralgleichung. Verhandlungen der deutschen physikalischen Gesellschaft 2 1900, S. 202.Google Scholar
  34. 34.
    Michael Rabin. Probabilistic automata. Information and Control, 1963, vol. 6, p. 230–245.CrossRefGoogle Scholar
  35. 35.
    Michael Rabin. Probabilistic algorithms. In Algorithms and Complexity, Recent Results and New Directions, J.F. Traub, Ed., Academic Press, NY, 1976, p. 21–39.Google Scholar
  36. 36.
    Friedrich Roesler. Riemann hypothesis as an eigenvalue problem. Linear Algebra Appl. 1986, vol. 81, p. 153–198.MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Peter Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proc. 35th FOCS, 1994, p. 124–134.Google Scholar
  38. 38.
    R. Solovay and V. Strassen. A fast Monte-Carlo test for primality. SIAM J. Comput, vol. 6, No. 1 (March), 1977, p. 84–85. See also SIAM J. Comput., vol. 7, No. 1 (Feb.), 1978, p. 118.MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    André Weil. Number theory and algebraic geometry. Proc. Intern. Congr. Math., 1950, Cambridge, vol. 2, p. 90–100.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • RŪsiņš Freivalds
    • 1
  1. 1.Department of Computer ScienceUniversity of LatviaRigaLatvia

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