Volume 11, 2005, Number 4

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.

Fermat numbers (F_n = 2²ⁿ + 1) and Mersenne numbers (M_m = 2^m − 1), m odd, are compared on the basis of integer structure, using the modular rings Z₄ and Z₆. The two numbers fall in different classes and this results in different composite row structures and different potentials for the formation of primes. The constraints on 2ⁿ and the right end digits for F_n result in fewer numbers over a given range than those for M_m. This is shown with two functions, which link the two numbers and show that F_n = (2^{x2 + y2 − 1} + 1): for primes y = n, but when n > 4, y ≠ n.