On the structure of certain counting polynomials

Robert J. Valenza
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 11, 2005, Number 4, Pages 1—11
Download full paper: PDF, 135 Kb

Details

Authors and affiliations

Robert J. Valenza
Claremont McKenna College

Abstract

We consider a natural generalization ζ(k)(s) = Σαn/ns (k ≥ 2) of the Riemann zeta function that arises from a modification of its classical Euler product expansion, for the most part here concentrating on the case k = 2. The associated coefficients αn correspond to a counting problem that may be addressed via a family of multivariable generating functions. Examples computed via symbolic manipulation suggest a recursive structure for these functions, which we prove. With this result in hand, the calculation of the αn may be facilitated by a more efficient, double modular algorithm, as worked out in a detailed example. We conclude with some observations and questions for the case k > 2.

Keywords

  • Riemann zeta function
  • Euler product
  • Multivariable generating functions
  • Symbolic manipulation algorithms

References

  1. Ivić, Aleksandar. The Riemann Zeta-Function, Wiley-Interscience, New York, 1985.
  2. Janusz, Gerald J. Algebraic Number Fields, Academic Press, New York, 1973.
  3. Ramakrishnan, Dinakar and Robert J. Valenza. Fourier Analysis on Number Fields, Springer-Verlag Graduate Texts in Mathematics, New York, 1999.
  4. Tucker, Alan. Applied Combinatorics (Second Edition), John Wiley & Sons, New York, 1984.

Related papers

Cite this paper

APA

Valenza, R. J. (2005). On the structure of certain counting polynomials. Notes on Number Theory and Discrete Mathematics, 11(4), 1-11.

Chicago

Valenza, Robert J. “On the Structure of Certain Counting Polynomials.” Notes on Number Theory and Discrete Mathematics 11, no. 4 (2005): 1-11.

MLA

Valenza, Robert J. “On the Structure of Certain Counting Polynomials.” Notes on Number Theory and Discrete Mathematics 11.4 (2005): 1-11. Print.

Comments are closed.