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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.
- Riemann zeta function
- Euler product
- Multivariable generating functions
- Symbolic manipulation algorithms
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Cite this paperAPA
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.