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Summation and the Poisson formula

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Abstract

By giving the definition of the sum of a series indexed by a set on which a group operates, we prove that the sum of the series that defines the Riemann zeta function, the Epstein zeta function and a few other series indexed by \({\mathbb {Z}}^k\) has an intrinsic meaning as a complex number, independent of the requirements of analytic continuation. The definition of the sum requires nothing more than algebra and the concept of absolute convergence. The analytical significance of the algebraically defined sum is then explained by an argument that relies on the Poisson formula for tempered distributions.

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Notes

  1. Condition (C) in Sect. 1.3 of Hardy’s book “Divergent Series” can be thought of Axiom (ii) for \(\Gamma ={\mathbb {Z}}\) and \(\gamma '=1\); Hardy’s (A) and (B) are Axiom (i).

  2. J-F Burnol’s paper [4] has this form of the Poisson formula on p. 798; on p. 799, he refers to the work of Connes [6] and that of Muntz explained by Titchmarsh in §2.11 [19].

  3. Akshay Venkatesh points out that Kudla and Rallis have more sophisticated versions and applications of translation-invariance in [10].

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Acknowledgements

The author thanks David Mumford for introducing him to the space \({\mathrm {C}}^{\infty }(X\times X'//\Gamma \times \Gamma ')\) in a beautiful course of lectures in 1979. He also thanks C. Kenig, J. Lagarias, W. Li, S. Miller, R. Narasimhan, W. Schlag, W. Schmid and D. Thakur for useful comments and assistance with references. The author also expresses his profound gratitude to the reviewer for going through the paper carefully and making several recommendations to improve its readability.

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  1. The University of Chicago, 5734 S. University Ave Il, Chicago, 60637, USA

    Madhav V. Nori

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Correspondence to Madhav V. Nori.

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Nori, M.V. Summation and the Poisson formula. Res Math Sci 7, 5 (2020). https://doi.org/10.1007/s40687-020-0204-2

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