A comprehensive study of the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}\exp{(-an^{N}x)}}{1-\exp{(-n^{N}x)}}, 0<a\leq 1,\ x>0$, $N\in\mathbb{N}$ and $h\in\mathbb{Z}$, is undertaken. Two of the general transformations of this series that we obtain here lead to two-parameter generalizations of Ramanujan's famous formula for $ζ(2m+1)$, $m>0$ and the transformation formula for $\logη(z)$. Numerous important special cases of our transformations are derived. An identity relating $ζ(2N+1), ζ(4N+1),\cdots, ζ(2Nm+1)$ is obtained for $N$ odd and $m\in\mathbb{N}$. Certain transcendence results of Zudilin- and Rivoal-type are obtained for odd zeta values and generalized Lambert series. A criterion for transcendence of $ζ(2m+1)$ and a Zudilin-type result on irrationality of Euler's constant $γ$ are also given. New results analogous to those of Ramanujan and Klusch for $N$ even, and a transcendence result involving $ζ\left(2m+1-\frac{1}{N}\right)$, are obtained.
Generalized Lambert series and arithmetic nature of odd zeta values
本文指出，Kanemitsu，Tanigawa和Yoshimoto研究的广义Lambert级数$\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}}{e^{n^{N}x}-1}$可以在Ramanujan's Lost Notebook的第$332$页上以稍微更一般的形式找到。我们延长了一个重要的变换，这一系列的Kanemitsu，谷川和Yoshimoto通过删除限制参数$N$和$h$，他们施加。从我们的扩展，我们推导出一个美丽的新的推广Ramanujan的著名公式奇zeta值，$N$奇和$m>0$，给出了$ζ(2m+1)$和$ζ(2Nm+1)$之间的关系。对于任何偶数$N$和$m\in\mathbb{Z}$，得到了与上述推广互补的结果。它推广了Wigert的一个变换，并可看作是$ζ\left(2m+1-\frac{1}{N}\right)$的一个公式.这些变换的应用包括推广的对数的Dedekind eta-函数$η(z)$，Zudilin和Rival型结果超越某些值，超越标准欧拉常数$γ$的转换。
Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$ . Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$ . Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$ . We also concl…
