Functional equations generating space-densifying curves

On Some Solutions of a Functional Equation Related to the Partial Sums of the Riemann Zeta Function

“…Thus, since G n (z) is an analytic function and the real part of its zeros is bounded [5], it is clear that b (j) n tends to infinity when j goes to infinity. On the contrary, in the case of the zeros of G n (z) were bounded, there would exist a convergent sequence of such zeros and it would be G n (z) ≡ 0 because of the Identity Principle [1,Theorem 1.3.7].…”

“…But, this property is deduced from [5,Theorem 7] where the reader can find a proof concerning the fact that the functions f β…”

“…Concretely, the cases n = 2 and n = 3 were used for modeling certain processes related to combustion of hydrogen in a car engine [2,5].…”

“…Some analytic properties of the solutions of the functional equation (1.1) F (z) + F (2z) + · · · + F (nz) = 0, z ∈ C, for each n ≥ 2, were studied in [5]. In fact, it was proved in [5,Theorem 7] that the functions f βn (z) ≡ z βn , with β n = a n + ib n belonging to the set of the zeros of (1.2) G n (z) ≡ 1 + 2 z + · · · + n z , are solutions of (1.1).…”

“…In fact, it was proved in [5,Theorem 7] that the functions f βn (z) ≡ z βn , with β n = a n + ib n belonging to the set of the zeros of (1.2) G n (z) ≡ 1 + 2 z + · · · + n z , are solutions of (1.1).…”