arXiv 1004.21682010-04-13math.NTmath.CO

狄利克雷级数$L_m(s)$的Generating Function

The Generating Function for the Dirichlet Series $L_m(s)$

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

备注：: 18 pages

摘要：

狄利克雷级数$L_m(s)$在数论中具有重要意义。Shanks定义了与这些狄利克雷级数有关的广义欧拉数和类数，记为$\{s_{m,n}\}_{n\geq 0}$。我们得到了$s_{m,n}$的指数母函数$s_m(x)$的一个公式，其中m是任意的正整数.特别地，对于m>1，比如说$m=bu^2$，其中b是无平方和u>1，我们证明了$s_m(x)$可以表示为四个函数$w(b,t)\sec (btx)(\pm \cos ((b-p)tx)\pm \sin (ptx))$的线性组合，其中p是满足$0\leq p\leq b$，$t|u^2$和$w(b,t)=K_bt/u$的整数，$K_b$是依赖于b的常数。此外，狄利克雷级数$L_m(s)$可以很容易地从生成函数公式为$s_m(x)$计算。最后，我们表明，在$s_{m,n}$公式中的主要成分有一个组合的解释，在由Ehrenborg和Readdy定义的m-签署的排列。原则上，这回答了Shanks提出的关于数字$s_{m,n}$的组合解释的问题。

英文摘要：

The Dirichlet series $L_m(s)$ are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with these 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 prove 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 an integer satisfying $0\leq p\leq b$, $t|u^2$ 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 m-signed permutations defined by Ehrenborg and Readdy. In principle, this answers a question posed by Shanks concerning a combinatorial interpretation for the numbers $s_{m,n}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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