The generating function for the Dirichlet series

Authors:
William Y. C. Chen, Neil J. Y. Fan and Jeffrey Y. T. Jia

Journal:
Math. Comp. **81** (2012), 1005-1023

MSC (2010):
Primary 11B68, 05A05

Published electronically:
July 26, 2011

MathSciNet review:
2869047

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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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Additional Information

**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

DOI:
https://doi.org/10.1090/S0025-5718-2011-02520-2

Keywords:
Dirichlet series,
generalized Euler and class number,
$Λ$-alternating augmented $m$-signed permutation,
$r$-cubical lattice,
Springer number

Received by editor(s):
April 13, 2010

Received by editor(s) in revised form:
December 26, 2010

Published electronically:
July 26, 2011

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.