Euler’s ϕ-Function

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§ I.1 Elementary inequalities for ϕ

  1. 1)

    http://static-content.springer.com/image/chp%3A10.1007%2F1-4020-3658-2_1/1-4020-3658-2_1_IEq183_HTML.gif for n ≠ 2 and n ≠ 6

    A.M. Vaidya. An inequality for Euler’s totient function. Math. Student 35 (1967), 79–80.

     
  2. 2)

    ϕ(n)>n2/3 for n>30

    D.G. Kendall and R. Osborn. Two simple lower bounds for Euler’s function. Texas J. Sci. 17 (1965), No. 3.

     
  3. 3)

    If a>6 and n>2, then

    http://static-content.springer.com/image/chp%3A10.1007%2F1-4020-3658-2_1/1-4020-3658-2_1_Equa_HTML.gif

    R.L. Goldstein. An inequality for Euler’s function ϕ(n). Math. Mag. 40 (1956), 131.

     
  4. 4)

    http://static-content.springer.com/image/chp%3A10.1007%2F1-4020-3658-2_1/1-4020-3658-2_1_IEq184_HTML.gif if n is composite

    W. Sierpiński. Elementary theory of numbers. Warsawa, 1964.

     
  5. 5)

    http://static-content.springer.com/image/chp%3A10.1007%2F1-4020-3658-2_1/1-4020-3658-2_1_IEq1_HTML.gif for n≥3

    H. Hatalová and T. Šalát. Remarks on two results in the elementary theory of numbers. Acta Fac. Rer. Natur Univ. Comenian. Math. 20 (1969), 113–117.

     

§ I.2 Inequalities for ϕ(mn)

  1. 1)

    ϕ(m) ϕ(n)≤ϕ(mn)≤n · ϕ(m); m, n=1, 2, 3, … (Simple consequence of the formula expressing ϕ)

     
  2. 2)

    (ϕ(mn))2≤ϕ(m2) · ϕ(n2); m, n=1, 2, 3, …

    T. Popoviciu. Gaz. Mat. (Bucureşti), 46 (1940), p. 334.

     

§ I.3 Relations connecting ϕ, σ, d

  1. 1)

    1. a)

      τ(n)≥ϕ(n)+d(n), n=2, 3, …

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