Robert J. Valenza

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 11, 2005, Number 4, Pages 1—11

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## Details

### Authors and affiliations

Robert J. Valenza

*Claremont McKenna College*

### Abstract

We consider a natural generalization ζ^{(k)}(*s*) = Σ*α _{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*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}_{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

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

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## Cite this paper

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

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

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