A Totient Function Inequality

  • Authors

    • N. Carella CUNY, New York
    https://doi.org/10.14419/ijams.v1i4.1117
  • Totient function, Prime numbers, Nicolas inequality, highly composite numbers, and Riemann hypothesis.
  • Abstract

    A new unconditional inequality of the totient function is contributed to the literature. This result is associated with various unsolved problems about the distribution of prime numbers.

     

     

  • References

    1. American Institute of Mathematics, aimath.org.
    2. L. Alaoglu and P. Erdos, on highly composite and similar numbers, Trans. Amer: Math. Soc. 56 (1944), 448-469.
    3. Briggs, K. Abundant numbers and the Riemann hypothesis. Experiment. Math. 15 (2006), no. 2, 251-256.
    4. William D. Banks, Derrick N. Hart, Pieter Moree, C. Wesley Nevans, The Nicolas and Robin inequalities with sums of two squares, arXiv: 0710.2424.
    5. Diamond, H. G.; Pintz, J. (2009). "Oscillation of Mertens' product formula". J. de Theorie des Nombres de Bordeaux 21: 523–533.
    6. G. H. Hardy and E. M. Wright, an Introduction to the Theory of Numbers, 5th ed., Oxford University Press, Oxford, 1979.
    7. Ivić, Aleksandar the Riemann zeta-function. The theory of the Riemann zeta-function with applications. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. ISBN: 0-471-80634-X.
    8. Tadej Kotnik, Computational Investigation of Five Problems in Analytic Number Theory, Preprint, 2006 available at www.
    9. Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, arXiv:math/0008177.
    10. Montgomery, Hugh L.; Vaughan, Robert C. Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics, 97. Cambridge University Press, Cambridge, 2007.
    11. S. Ramanujan, Highly composite numbers, Proc. London Math. Soc. 14 (1915), 347407.
    12. J.-L. Nicolas, Petites valeurs de la fonction d’Euler, J. Number Theory 17 (1983) 375–388.
    13. G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypoth`ese de Riemann, J. Math. Pures Appl. (9) 63 (1984) 187–213.
    14. Y.-J. Choie, N. Lichiardopol, P. Moree, P. Sole, On Robin's criterion for the Riemann Hypothesis, arXiv:math/0604314.
    15. Shapiro, Harold N. Introduction to the theory of numbers. Pure and Applied Mathematics. A Wiley-Interscience Publication. New York, 1983.
    16. G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics 46, (Cambridge University Press, Cambridge, 1995.
    17. Mark B. Villarino, Mertens' Proof of Mertens' Theorem, arXiv:math/0504289v3.
    18. Wikipedia.org.
    19. Weingartner, Andreas. The distribution functions of $sigma (n)/n$ and $n/phi (n) $. Proc. Amer. Math. Soc. 135 (2007), no. 9, 2677-2681.
    20. Weingartner, Andreas. The limiting distribution of the divisor function. J. Number Theory 129 (2009), no. 6, 1432-1442.
    21. Marek Wójtowicz, Robin's inequality and the Riemann hypothesis, Proc. Japan Acad. Ser. A Math. Sci. Volume 83, Number 4 (2007), 47-49.
  • Downloads

  • How to Cite

    Carella, N. (2013). A Totient Function Inequality. International Journal of Advanced Mathematical Sciences, 1(4), 185-189. https://doi.org/10.14419/ijams.v1i4.1117