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
Full paper (PDF, 135 Kb)

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Robert J. Valenza
Claremont McKenna College

Abstract

We consider a natural generalization ζ^(k)(s) = Σα_n/n^s (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 α_n 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 α_n 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.