ON SOME SOLUTIONS OF A FUNCTIONAL EQUATION RELATED TO THE PARTIAL SUMS OF THE RIEMANN ZETA FUNCTION

Title & Authors
ON SOME SOLUTIONS OF A FUNCTIONAL EQUATION RELATED TO THE PARTIAL SUMS OF THE RIEMANN ZETA FUNCTION
Martinez, Juan Matias Sepulcre;

Abstract
In this paper, we prove that infinite-dimensional vector spaces of -dense curves are generated by means of the functional equations f(x)+f(2x)+$\small{{\cdots}}$+f(nx) = 0, with $\small{n{\geq}2}$, which are related to the partial sums of the Riemann zeta function. These curves $\small{{\alpha}}$-densify a large class of compact sets of the plane for arbitrary small $\small{{\alpha}}$, 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 $\small{n^{th}}$ 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;
Language
English
Cited by
1.
On the Analytic Solutions of the Functional Equations w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0, Mediterranean Journal of Mathematics, 2015, 12, 3, 667
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