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On multiple analogues of Ramanujan’s formulas for certain Dirichlet series. (English) Zbl 1211.11100

Author’s summary: “The author proves multiple analogues of the famous Ramanujan formulas for certain Dirichlet series which were introduced in his well-known notebooks as well as some multiple versions of analogous formulas of Ramanujan which were given by Berndt and so on.”
The main theorem is as follows: for \(r\in\mathbb N\) with \(r\geq 2\) and \(p_ 1,p_ 2,\dots,p_ {r-1}\in\mathbb N\), \[ \mathcal X_ r(2p_ 1,\dots,2p_ {r-1},s)=(-1)^ {r-1}\mathcal X_ 1\left(s+2\sum_ {j=1}^ {r-1}p_ j\right) \] holds for \(\text{Re}\, s>1\), and that \(\mathcal X_ r(2p_ 1,\dots,2p_ {r-1},s)\) can be continued meromorphically to \(\mathbb C\) by this equation.
Here \(r\)-ple analogues of \(\mathcal X_1(s)=\sum_{k=1}^\infty \frac{\coth(k\pi)}{k^s}\) are considered, namely
\[ \mathcal X_ r(2p_ 1,\dots,2p_ {r-1},s):=\sum_{k_j}\frac{\coth(k\pi)}{\prod_{d=1}^{r-1}(\sum_{j=1}^d k_j)^{2p_d}\cdot(\sum_{j=1}^r k_j)^s}. \]

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
11M41 Other Dirichlet series and zeta functions
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References:

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