The Fourier Transform of the Non-Trivial Zeros of the Zeta Function
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Keywords

Zeta function, non-trivial zeros, Fourier transform, series.

How to Cite

Levente Csoka. (2017). The Fourier Transform of the Non-Trivial Zeros of the Zeta Function. Journal of Advances in Applied &Amp; Computational Mathematics, 4(1), 23–25. https://doi.org/10.15377/2409-5761.2017.04.4

Abstract

 The non-trivial zeros of the Riemann zeta function and the prime numbers can be plotted by a modified von Mangoldt function. The series of non-trivial zeta zeros and prime numbers can be given explicitly by superposition of harmonic waves. The Fourier transform of the modified von Mangoldt functions shows interesting connections between the series. The Hilbert-Pólya conjecture predicts that the Riemann hypothesis is true, because the zeros of the zeta function correspond to eigen values of a positive operator and this idea encouraged to investigate the eigenvalues itself in a series. The Fourier transform computations is verifying the Riemann hypothesis and give evidence for additional conjecture that those zeros and prime numbers arranged in series that lie in the critical, 1/2. positive upper half plane and over the positive integers, respectively.
https://doi.org/10.15377/2409-5761.2017.04.4
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References

J. Brian Conrey The Riemann Hypothesis, Notices of the AMS 2003 50(3): 341-353.

Titchmarsh EC. The Theory of the Riemann Zeta-Function, second edition, edited and with a preface by D. R. Heath- Brown, The Clarendon Press, Oxford University Press, New York, 1986.

Biane P, Pitman JM. Yor Probability laws related to the Jacobi theta and Riemann zeta functions and Brownian Excursion Bull. Amer. Math. Soc. (N.S.) 2001; 38: 435-65.

E. Bombieri Problems of the Millennium: The Riemann Hypothesis, Institute for Advanced Study, Princeton NJ. 08540; 1-11.

Katz NM and Sarnak Random matrices P. Frobenius eigenvalues and monodromy, American Mathematics Society, Providence RI 1999.