The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era,...The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.展开更多
The finite field F<sub>q</sub> has q elements, where q = p<sup>k</sup> for prime p and k∈N. Then F<sub>q</sub>[x] is a unique factorization domain and its polynomials can be b...The finite field F<sub>q</sub> has q elements, where q = p<sup>k</sup> for prime p and k∈N. Then F<sub>q</sub>[x] is a unique factorization domain and its polynomials can be bijectively associated with their unique (up to order) factorizations into irreducibles. Such a factorization for a polynomial of degree n can be viewed as conforming to a specific template if we agree that factors with higher degree will be written before those with lower degree, and factors of equal degree can be written in any order. For example, a polynomial f(x) of degree n may factor into irreducibles and be written as (a)(b)(c), where deg a ≥ deg b ≥deg c. Clearly, the various partitions of n correspond to the templates available for these canonical factorizations and we identify the templates with the possible partitions. So if f(x) is itself irreducible over F<sub>q</sub>, it would belong to the template [n], and if f(x) split over F<sub>q</sub>, it would belong to the template [n] Our goal is to calculate the cardinalities of the sets of polynomials corresponding to available templates for general q and n. With this information, we characterize the associated probabilities that a randomly selected member of F<sub>q</sub>[x] belongs to a given template. Software to facilitate the investigation of various cases is available upon request from the authors.展开更多
文摘The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.
文摘The finite field F<sub>q</sub> has q elements, where q = p<sup>k</sup> for prime p and k∈N. Then F<sub>q</sub>[x] is a unique factorization domain and its polynomials can be bijectively associated with their unique (up to order) factorizations into irreducibles. Such a factorization for a polynomial of degree n can be viewed as conforming to a specific template if we agree that factors with higher degree will be written before those with lower degree, and factors of equal degree can be written in any order. For example, a polynomial f(x) of degree n may factor into irreducibles and be written as (a)(b)(c), where deg a ≥ deg b ≥deg c. Clearly, the various partitions of n correspond to the templates available for these canonical factorizations and we identify the templates with the possible partitions. So if f(x) is itself irreducible over F<sub>q</sub>, it would belong to the template [n], and if f(x) split over F<sub>q</sub>, it would belong to the template [n] Our goal is to calculate the cardinalities of the sets of polynomials corresponding to available templates for general q and n. With this information, we characterize the associated probabilities that a randomly selected member of F<sub>q</sub>[x] belongs to a given template. Software to facilitate the investigation of various cases is available upon request from the authors.
