Fourier transforms with only real zeros

Newman, Charles M.

doi:10.1090/S0002-9939-1976-0434982-5

Fourier transforms with only real zeros
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by Charles M. Newman

Proc. Amer. Math. Soc. 61 (1976), 245-251

DOI: https://doi.org/10.1090/S0002-9939-1976-0434982-5

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Abstract:

The class of even, nonnegative, finite measures $\rho$ on the real line such that for any $b > 0$ the Fourier transform of $\exp ( - b{t^2})d\rho (t)$ has only real zeros is completely determined. This result is then applied to the Riemann hypothesis.

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