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

  • Martinez, Juan Matias Sepulcre
  • Received : 2012.02.28
  • Published : 2014.01.31

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)+${\cdots}$+f(nx) = 0, with $n{\geq}2$, which are related to the partial sums of the Riemann zeta function. These curves ${\alpha}$-densify a large class of compact sets of the plane for arbitrary 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 $n^{th}$ 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

References

  1. R. B. Ash, Complex Variables, Academic Press, New York, 1971.
  2. G. Mora, A note on the functional equation F(z) + F(2z) + ${\ldots}$ + F(nz) = 0, J. Math. Anal. Appl. 340 (2008), no. 1, 466-475. https://doi.org/10.1016/j.jmaa.2007.08.045
  3. G. Mora and Y. Cherrault, Characterization and generation of ${\alpha}$-dense curves, Comput. Math. Appl. 33 (1997), no. 9, 83-91.
  4. G. Mora and Y. Cherrault, A new approach to the reduction of multiple integrals to simple ones using Chebyshev's kernels, Kybernetes 37 (2008), no. 1, 104-119. https://doi.org/10.1108/03684920810851023
  5. G. Mora, Y. Cherrault, and A. Ziadi, Functional equations generating space-densifying curves, Comput. Math. Appl. 39 (2000), no. 9-10, 45-55.
  6. G. Mora and J. M. Sepulcre, On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function, J. Math. Anal. Appl. 350 (2009), no. 1, 409-415. https://doi.org/10.1016/j.jmaa.2008.09.068
  7. G. Mora and J. M. Sepulcre, The critical strips on the sums 1+$2^z$+${\ldots}$+$n^z$, Abstr. Appl. Anal. 2011 (2011), Art. ID 909674, 15 pp. doi:10.1155/2011/909674. https://doi.org/10.1155/2011/909674
  8. H. Sagan, Space-Filling Curves, Springer-Verlag, New York, 1994.

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 vol.12, pp.3, 2015, https://doi.org/10.1007/s00009-014-0444-8