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\(L\)-functions as distributions

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Abstract

We define an axiomatic class of \(L\)-functions extending the Selberg class. We show in particular that one can recast the traditional conditions of an Euler product, analytic continuation and functional equation in terms of distributional identities akin to Weil’s explicit formula. The generality of our approach enables some new applications; for instance, we show that the \(L\)-function of any cuspidal automorphic representation of \({{\mathrm{GL}}}_3(\mathbb {A}_\mathbb {Q})\) has infinitely many zeros of odd order.

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Notes

  1. In full generality it is known in a slightly weaker form that includes the prime powers in the sum, and those may be removed under a mild hypothesis; see [1, 25].

  2. Selberg acknowledges in a footnote of his paper that we may take \(\lambda \) to be a half-integer in every known case. It seems likely that he did not intend for his definition to be taken as a serious attempt at generalization, but rather as a recognition of the fact that the \(\Gamma \)-factors are non-canonical because of the Legendre and Gauss multiplication formulas. We note that those identities have analogues at the finite places as well, e.g. the Legendre duplication formula is analogous to the “difference of squares” identity \(1-p^{-2s}=(1-p^{-s})(1+p^{-s})\), but this ambiguity causes no real confusion. Note also that the analogue of \(\Gamma (\lambda {s})\) is the generalized Dirichlet series \(1/(1-p^{-2\lambda {s}})\), which is not permitted under Selberg’s definition unless \(\lambda \) is a half-integer.

  3. It is tempting to consider more general singularities as well, but we quickly find ourselves in a much larger landscape of functions that is presumably very hard to classify; for instance, the Selberg zeta-functions and their trace formulae give identities of this type with second-order singularities.

  4. There is also an argument if favor of keeping examples like \(\zeta (2s-\frac{1}{2})\) in the definition: Shimura’s integral representation for the symmetric square \(L\)-function could be viewed as an extension of the Rankin–Selberg method to this example, and that in turn was a key ingredient in the proof of the Gelbart–Jacquet lift.

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Acknowledgments

This work was carried out during a year-long stay at the Research Institute for Mathematical Sciences, Kyoto, Japan. It is a pleasure to thank all of the RIMS staff, in particular my host, Akio Tamagawa, for their generous hospitality. I would also like to thank Akihiko Yukie for organizing the Conference on Automorphic Forms at Kyoto University in June 2013, which provided the impetus for this work. Finally, I thank Frank Thorne for performing some computations in relation to Remarks 1.12, and Brian Conrey, David Farmer, Peter Sarnak and Akshay Venkatesh for helpful suggestions.

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Correspondence to Andrew R. Booker.

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The author was supported by EPSRC Fellowship EP/H005188/1.

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Booker, A.R. \(L\)-functions as distributions. Math. Ann. 363, 423–454 (2015). https://doi.org/10.1007/s00208-015-1178-z

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