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Zbl 1211.11100
Tsumura, Hirofumi
On multiple analogues of Ramanujan's formulas for certain Dirichlet series. (English)
[J] J. Théor. Nombres Bordx. 20, No. 1, 219-226 (2008). ISSN 1246-7405

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 $r\in\Bbb N$ with $r\geq 2$ and $p\sb 1,p\sb 2,\dots,p\sb {r-1}\in\Bbb N$, $$\cal X\sb r(2p\sb 1,\dots,2p\sb {r-1},s)=(-1)\sp {r-1}\cal X\sb 1\left(s+2\sum\sb {j=1}\sp {r-1}p\sb j\right)$$ holds for $\text{Re}\, s>1$, and that $\cal X\sb r(2p\sb 1,\dots,2p\sb {r-1},s)$ can be continued meromorphically to $\Bbb C$ by this equation.
[Olaf Ninnemann (Berlin)]

MSC 2000:: *11M32
11M41 Other Dirichlet series and zeta functions

Keywords: multiple zeta values

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