Abstract: The Dirichlet series are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with the Dirichlet series, denoted by . We obtain a formula for the exponential generating function of , where is an arbitrary positive integer. In particular, for , say, , where is square-free and , we show that can be expressed as a linear combination of the four functions , where is a nonnegative integer not exceeding , and with being a constant depending on . Moreover, the Dirichlet series can be easily computed from the generating function formula for . Finally, we show that the main ingredient in the formula for has a combinatorial interpretation in terms of the -alternating augmented -signed permutations defined by Ehrenborg and Readdy. More precisely, when is square-free, this answers a question posed by Shanks concerning a combinatorial interpretation of the numbers . When is not square-free, say , the numbers can be written as a linear combination of the numbers of -alternating augmented -signed permutations with integer coefficients, where .
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William Y. C. Chen Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email:
chen@nankai.edu.cn
Neil J. Y. Fan Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email:
fjy@cfc.nankai.edu.cn
Jeffrey Y. T. Jia Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email:
jyt@cfc.nankai.edu.cn