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

Abstract

Abstract. In this paper, we prove that infinite-dimensional vector spaces of α-dense curves are generated by means of the functional equations f (x) + f (2x) + · · · + f (nx) = 0, with n ≥ 2, which are related to the partial sums of the Riemann zeta function. These curves α-densify a large class of compact sets of the plane for arbitrary small α, extending the known result that this holds for the cases n = 2, 3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the n th power of the density approaches the Jordan content of the compact set which the curve densifies.