Counterexamples to a conjecture by Gun and co-workers, its correct reformulation and the transcendence of some series

Details Counterexamples to a conjecture by Gun and co-workers, its correct reformulation and the transcendence of some series Counterexamples to a conjecture by Gun and co-workers, its correct reformulation and the transcendence of some series

2009-11-12

arxiv.org/abs/0911.2441v4

Mathematics

Number Theory

13 pages, no figures. Some minor revisions. Title changed. Submitted to "IJMMS" (04/06/2011)

Scientific paper

In a recent work, Gun, Murty and Rath have introduced a 'theorem' asserting that $\sum_{n=-\infty}^{\infty}{(n+\alpha)^{-k}}\,$ yields a transcendental number for all $\,\alpha \in \mathbb{Q} \, \backslash \, \mathbb{Z}\,$, $k$ being an integer greater than 1. I show here in this short paper that this conjecture is \emph{false} whenever $k$ is odd and $\alpha$ is a half-integer. I also prove that these are the only counterexamples, which allows for a correct reformulation. The resulting theorem implies the transcendence of both the polygamma function at rational entries and certain zeta series.

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