Generalized Lambert series, Raabe's integral and a two-parameter generalization of Ramanujan's formula for ζ(2m+1)

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dc.contributor.author Dixit, Atul
dc.contributor.author Gupta, Rajat
dc.contributor.author Kumar, Rahul
dc.contributor.author Maji, Bibekananda
dc.date.accessioned 2018-02-15T09:35:34Z
dc.date.available 2018-02-15T09:35:34Z
dc.date.issued 2018-01
dc.identifier.citation Dixit, Atul; Gupta, Rajat; Kumar, Rahul and Maji, Bibekananda, “Generalized Lambert series, Raabe's integral and a two-parameter generalization of Ramanujan's formula for ζ(2m+1)”, arXiv, Cornell University Library, DOI: arXiv:1801.09181, Jan. 2018. en_US
dc.identifier.uri https://repository.iitgn.ac.in/handle/123456789/3460
dc.identifier.uri http://arxiv.org/abs/1801.09181
dc.description.abstract A comprehensive study of the generalized Lambert series ∑n=1∞nN−2hexp(−anNx)1−exp(−nNx),0<a≤1, x>0, N∈N and h∈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),⋯,ζ(2Nm+1) is obtained for N odd and m∈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(2m+1−1N), are obtained. en_US
dc.description.statementofresponsibility by Atul Dixit, Rajat Gupta, Rahul Kumar and Bibekananda Maji
dc.language.iso en en_US
dc.publisher Cornell University Library en_US
dc.title Generalized Lambert series, Raabe's integral and a two-parameter generalization of Ramanujan's formula for ζ(2m+1) en_US
dc.type Preprint en_US


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