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Models of Computation, Riemann Hypothesis, and Classical Mathematics

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SOFSEM’ 98: Theory and Practice of Informatics (SOFSEM 1998)

Part of the book series: Lecture Notes in Computer Science ((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.

Research supported by Grant No.96.0282 from the Latvian Council of Science

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References

  1. Leonard Adleman. Molecular computation of solutions to combinatorial problems. Science, 1994, vol. 266, p. 1021–1024.

    Article  Google Scholar 

  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. N. C. Ankeny. The least quadratic non residue. Annals of Mathematics, 1952, vol. 55, p. 65–72.

    Article  MathSciNet  Google Scholar 

  4. Eric Bach. Fast algorithms under the Extended Riemann Hypothesis: a concrete estimate. Proc. 14th STOC, 1982, p. 290–295.

    Google Scholar 

  5. Eric Bach. Realistic analysis of some randomized algorithms. Journal of Computer and System Sciences, 1991, vol. 42, No. 1, p. 30–53.

    Article  MATH  MathSciNet  Google Scholar 

  6. Paul Benioff. Quantum mechanical Hamiltonian models of Turing machines. J. Statistical Physics, 1982, vol. 29, p. 515–546.

    Article  MATH  MathSciNet  Google Scholar 

  7. Michael Ben-Or. Probabilistic algorithms in finite fields. Proc. 22nd FOCS, 1981, p. 394–398.

    Google Scholar 

  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. Alan Cobham. The Recognition Problem for the Set of Perfect Squares. Proc. FOCS, 1966, p. 78–87.

    Google Scholar 

  10. Pierre Deligne. La conjeture de Weil. Publ. Math. Inst. HES, v. 43, 1974, p. 273–307.

    Article  MathSciNet  Google Scholar 

  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. Richard Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 1982, vol. 21, No. 6/7, p. 467–488.

    Article  MathSciNet  Google Scholar 

  13. Ernests Fogels. On the zeros of L-functions. Acta Arithmetica, 1965, vol. 11, p. 67–96.

    MATH  MathSciNet  Google Scholar 

  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. 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).

    MathSciNet  Google Scholar 

  16. RŪsiņš Freivalds. Fast probabilistic algorithms. Lecture Notes in Computer Science, 1979, vol. 74, p. 57–69.

    Google Scholar 

  17. RŪsiņš Freivalds. Probabilistic two-way machines. Lecture Notes in Computer Science, 1981, vol. 118, p. 33–45.

    Google Scholar 

  18. RŪsiņš Freivalds. Space and reversal complexity of probabilistic one-way Turing machines. Annals of Discrete Mathematics, 1985, vol. 24, p. 39–50.

    MathSciNet  Google Scholar 

  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. 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. 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. Ming-Deh A. Huang. Riemann Hypothesis and finding roots over finite fields. Proc. 17th STOC, 1985, p. 121–130.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. Lila Kari. DNA computers, tomorrow’s reality. Bulletin of the EATCS, vol. 59, p. 256–266.

    Google Scholar 

  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. Attila Kondacs and John Watrous. On the power of quantum finite state automata. In Proc. 38th FOCS, 1997, p. 66–75.

    Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  31. Hugh L. Montgomery. Topics in Multiplicative Number Theory. Lecture Notes in Mathematics, 1971. vol. 227.

    Google Scholar 

  32. Cristopher Moore, James P. Crutchfield Quantum automata and quantum grammars. Manuscript available at http://xxx.lanl.gov/abs/quant-ph/9707031

  33. Max Planck. Uber eine Verbesserung der Wien’schen Spectralgleichung. Verhandlungen der deutschen physikalischen Gesellschaft 2 1900, S. 202.

    Google Scholar 

  34. Michael Rabin. Probabilistic automata. Information and Control, 1963, vol. 6, p. 230–245.

    Article  Google Scholar 

  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. Friedrich Roesler. Riemann hypothesis as an eigenvalue problem. Linear Algebra Appl. 1986, vol. 81, p. 153–198.

    Article  MATH  MathSciNet  Google Scholar 

  37. Peter Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proc. 35th FOCS, 1994, p. 124–134.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  39. André Weil. Number theory and algebraic geometry. Proc. Intern. Congr. Math., 1950, Cambridge, vol. 2, p. 90–100.

    MathSciNet  Google Scholar 

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Freivalds, R. (1998). Models of Computation, Riemann Hypothesis, and Classical Mathematics. In: Rovan, B. (eds) SOFSEM’ 98: Theory and Practice of Informatics. SOFSEM 1998. Lecture Notes in Computer Science, vol 1521. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49477-4_6

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  • DOI: https://doi.org/10.1007/3-540-49477-4_6

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