Collatz Problem , the (3n + 1) conjecture , discrete dynamical systems , Dirichlet series , functional equations , analytic number theory
Abstract:
A series of relatively simple equivalences to the Collatz conjecture, concerning the Collatz mapping [...] are presented. [...] The main topic of this paper consists in investigating a certain linear equation in the space of special Dirichlet series. The conjecture that this equation possesses a null space of dimension 1, generated by the Riemann zeta function, is equivalent to the Collatz conjecture. A number of analytic properties of the operator, which defines the linear equation, is given, some of them concern problems of analytic continuation in the complex domain. A few remarks with respect to generalizations of those problems conclude the paper.
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