Generalized Lambert series and arithmetic nature of odd zeta values

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dc.contributor.author Dixit, Atul
dc.contributor.author Maji, Bibekananda
dc.date.accessioned 2019-06-19T11:12:53Z
dc.date.available 2019-06-19T11:12:53Z
dc.date.issued 2019-02
dc.identifier.citation Dixit, Atul and Maji, Bibekananda, “Generalized Lambert series and arithmetic nature of odd zeta values”, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, DOI: 10.1017/prm.2018.146, vol. 150, no. 2, pp, 741-769, Feb. 2019. en_US
dc.identifier.uri https://doi.org/10.1017/prm.2018.146
dc.identifier.uri https://repository.iitgn.ac.in/handle/123456789/4454
dc.description.abstract It is pointed out that the generalized Lambert series studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2m + 1) and ζ(2Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2m + 1 − 1/N). Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function η(z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ.
dc.description.statementofresponsibility by Atul Dixit and Bibekananda Maji
dc.format.extent vol. 150, no. 2, pp, 741-769
dc.language.iso en en_US
dc.publisher Cambridge University Press en_US
dc.subject Lambert series en_US
dc.subject Dedekind eta function en_US
dc.subject odd zeta values Euler's en_US
dc.subject constant Ramanujan's formula en_US
dc.subject transcendence en_US
dc.title Generalized Lambert series and arithmetic nature of odd zeta values en_US
dc.type Article en_US
dc.relation.journal Proceedings of the Royal Society of Edinburgh: Section A Mathematics


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