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There exist a "standard" or canonical way to construct a real valued function whose Mellin transform has a prescribed set of zeroes? Clearly for some set of zeroes this could be impossible but for the "admissible ones"?

Thank you in advance for any suggestion.

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  • $\begingroup$ Presumably the basic approach would be to construct a function with a certain set of zeroes, and then take an inverse Mellin transform. To make the inverse transform real-valued, the original function should have some other conjugate-symmetry property, but this seems possible to check for functions constructed via Weierstrass factorization. $\endgroup$
    – Will Sawin
    Commented Jun 5, 2022 at 12:15
  • $\begingroup$ The problem with this approach for me is that I can't guarantee that the function will be integrable along vertical lines in the complex plane. $\endgroup$
    – MathG
    Commented Jun 5, 2022 at 12:45

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