Three results on the distribution of arithmetic functions

Let ϕ(n) be the Euler totient function and σ(n) be the sum of divisors function. In this thesis, we discuss three results. In the first result, we will derive an upper bound for the number of preimages of the iterates of ϕ and σ. In the second result, we discuss the distribution of polynomially-defined additive functions: An integer-valued additive function f such that there exist a polynomial F(x) ∈ Z[x] with the condition f(p) = F(p) for all primes p. In the third result, we discuss the distribution in R/Z of the sequences {αϕ(n)}∞ n=1 and {ασ(n)}∞ n=1 whereαis an irrational number of "finite type".