Sepulcre, Juan Matias
On some solutions of a functional equation related to the partial sums of the Riemann zeta function
Bulletin of the Korean Mathematical Society. 2014, 51(1): 29-41. doi:10.4134/BKMS.2014.51.1.029
URI: http://hdl.handle.net/10045/35671
DOI: 10.4134/BKMS.2014.51.1.029
ISSN: 1015-8634 (Print)
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 nth power of the density approaches the Jordan content of the compact set which the curve densifies.
Keywords:Functional equations, Space-filling curves, Partial sums of the Riemann zeta function, Alpha-dense curves, Property of quadrature
Korean Mathematical Society
info:eu-repo/semantics/article