Fractal Patterns in Prime Numbers Distribution

  • Carlo Cattani
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6017)


One of the main tasks in the analysis of prime numbers distribution is to single out hidden rules and regular features like periodicity, typical patterns, trends, etc. The existence of fractal shapes, patterns and symmetries in prime numbers distribution are discussed.


Fractals fractal dimension random walk prime numbers Riemann hypothesis 


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  1. 1.
    Ares, S., Castro, M.: Hidden structure in the randomness of the prime number sequence? Physica A 360, 285–296 (2006)CrossRefGoogle Scholar
  2. 2.
    Arneado, A., Bacry, E., Graves, P.V., Muzy, J.F.: Characterizing long-range correlations in DNA sequences from wavelet analysis. Phys. Rev. Lett. 74, 3293–3296 (1995)CrossRefGoogle Scholar
  3. 3.
    Arneado, A., D’Aubenton-Carafa, Y., Audit, B., Bacry, E., Muzy, J.F., Thermes, C.: What can we learn with wavelets about DNA sequences? Physica A 249, 439–448 (1998)Google Scholar
  4. 4.
    Berger, J.A., Mitra, S.K., Carli, M., Neri, A.: Visualization and analysis of DNA sequences using DNA walks. Journal of The Franklin Institutes 341, 37–53 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bernaola-Galván, P., Román-Roldán, R., Oliver, J.L.: Compositional segmentation and long-range fractal correlations in DNA sequences. Phys. Rev. E 55(5), 5181–5189 (1996)CrossRefGoogle Scholar
  6. 6.
    Cattani, C.: Wavelet Algorithms for DNA Analysis. In: Elloumi, M., Zomaya, A.Y. (eds.) To appear on Algorithms in Computational Molecular Biology: Techniques, Approaches and Applications. Wiley Series in Bioinformatics, ch. 35. Wiley-Blackwell, Chichester (2010)Google Scholar
  7. 7.
    Coward, E.: Equivalence of two Fourier methods for biological sequences. Journal of Mathematical Biology 36, 64–70 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dodin, G., Vandergheynst, P., Levoir, P., Cordier, C., Marcourt, L.: Fourier and Wavelet Transform Analysis, a Tool for Visualizing Regular Patterns in DNA Sequences. J. Theor. Biol. 206, 323–326 (2000)CrossRefGoogle Scholar
  9. 9.
    Edwards, H.M.: Riemann’s zeta-function. Academic Press, London (1974)zbMATHGoogle Scholar
  10. 10.
    Herzel, H., Trifonov, E.N., Weiss, O., Grosse, I.: Interpreting correlations in biosequences. Physica A 249, 449–459 (1998)Google Scholar
  11. 11.
    Li, W., Kaneko, K.: Long-range correlations and partial 1/f α spectrum in a noncoding DNA sequence. Europhys. Lett. 17, 655–660 (1992)CrossRefGoogle Scholar
  12. 12.
    Littlewood, J.E.: Sur la distribution des nombres premieres. C. R. Acad. Sci. Paris 158, 1869–1872 (1914)zbMATHGoogle Scholar
  13. 13.
    Narkiewicz, W.: The development of prime number theory. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  14. 14.
    Peng, C.-K., Buldryev, S.V., Goldberg, A.L., Havlin, S., Sciortino, F., Simons, M., Stanley, H.E.: Long-range correlations in nucleotide sequences. Nature 356, 168–170 (1992)CrossRefGoogle Scholar
  15. 15.
    Peng, C.-K., Buldryev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberg, A.L.: Mosaic organization of DNA nucleotides. Phys. Rev. E 49, 1685–1689 (1994)CrossRefGoogle Scholar
  16. 16.
    Shapiro, H.H.: Introduction to the Theory of Numbers. John Wiley & Sons, New York (1983)zbMATHGoogle Scholar
  17. 17.
    Schlesinger, M.: On the Riemann hypothesis: a fractal random walk approach. Physica A 138, 310–319 (1986)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Voss, R.F.: Evolution of Long-Range Fractal Correlations and 1/f Noise in DNA Base Sequences. Physical Review Letters 68(25), 3805–3808 (1992)CrossRefGoogle Scholar
  19. 19.
    Weiss, O., Herzel, H.: Correlations in protein sequences and property codes. J. Theor. Biol. 190, 341–353 (1998)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Carlo Cattani
    • 1
  1. 1.diFarmaUniversity of SalernoFiscianoItaly

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