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The generating function for the Dirichlet series $ L_m(s)$


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
DOI: https://doi.org/10.1090/S0025-5718-2011-02520-2
Published electronically: July 26, 2011
MathSciNet review: 2869047
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Abstract: The Dirichlet series $ L_m(s)$ are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with the Dirichlet series, denoted by $ \{s_{m,n}\}_{n\geq 0}$. We obtain a formula for the exponential generating function $ s_m(x)$ of $ s_{m,n}$, where $ m$ is an arbitrary positive integer. In particular, for $ m>1$, say, $ m=bu^2$, where $ b$ is square-free and $ u>1$, we show that $ s_m(x)$ can be expressed as a linear combination of the four functions $ w(b,t)\sec (btx)(\pm \cos ((b-p)tx)\pm \sin (ptx))$, where $ p$ is a nonnegative integer not exceeding $ b$, $ t\vert u^2$ and $ w(b,t)=K_bt/u$ with $ K_b$ being a constant depending on $ b$. Moreover, the Dirichlet series $ L_m(s)$ can be easily computed from the generating function formula for $ s_m(x)$. Finally, we show that the main ingredient in the formula for $ s_{m,n}$ has a combinatorial interpretation in terms of the $ \Lambda$-alternating augmented $ m$-signed permutations defined by Ehrenborg and Readdy. More precisely, when $ m$ is square-free, this answers a question posed by Shanks concerning a combinatorial interpretation of the numbers $ s_{m,n}$. When $ m$ is not square-free, say $ m=bu^2$, the numbers $ K_b^{-1}s_{m,n}$ can be written as a linear combination of the numbers of $ \Lambda$-alternating augmented $ bt$-signed permutations with integer coefficients, where $ t\vert u^2$.


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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.

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