Abstract
We utilize a combination of integral transforms, including the Laplace transform, with some classical results in analytic number theory concerning the Riemann ξ-function, to obtain a new integral equation. This integral equation is generalized to self-dual principal automorphic L-functions. We also provide a new proof of known functional-type identities from analytic number theory, and recast some criteria associated with the RH. An application of our integral equation to the Dirichlet problem in the half plane is stated, giving a new application of the Riemann ξ-function integral.
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Acknowledgement
We thank a referee for suggesting to improve the main result by extending it to self-dual automorphic L-functions.
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Patkowski, A.E. A new integral equation and integrals associated with number theory. Anal Math 47, 881–892 (2021). https://doi.org/10.1007/s10476-021-0076-8
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DOI: https://doi.org/10.1007/s10476-021-0076-8