Arithmetic properties of polynomial specializations over finite fields
In this paper, we look at various arithmetic properties of the set of those positive integers n whose sum of digits in a fixed base b > 1 is a fixed positive integer s. For example, we prove that such integers can have many prime factors, that they are not very smooth, and that most such integers have a large prime factor dividing the value of their Euler φ function.
For k = 1,2,... let denote the harmonic number . In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have , , and for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and .
The classical system of functional equations (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to (n ∈ ℕ) with complex valued sequences . This leads to new results on the periodic integrable and the aperiodic continuous solutions F:ℝ₊ → ℂ interrelating the theory of functional equations and the theory of arithmetic functions.