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Approximation by Riemann zeta zeros

[edit]

{wb:Fundamentals of Physics}

The real part of the sum over the zeta zeros:

-\sum _{i=1}^{\infty }n^{\rho (i)}

, where

ρ (i)

is the

i

-th zeta zero, peaks at primes, as can be seen in the adjoining graph, and can also be verified through numerical computation. It does not sum up to the Von Mangoldt function.^[1]

The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to imaginary part of the Riemann zeta function zeros. This is sometimes called a duality.

The Fourier transform of the von Mangoldt function gives a spectrum with imaginary parts of Riemann zeta zeros as spikes at the $x$ -axis ordinates (right), while the von Mangoldt function can be approximated by zeta zero waves (left)

References

[edit]

^ Conrey, J. Brian (March 2003). "The Riemann hypothesis" (PDF). Notices Am. Math. Soc. 50 (3): 341–353. Zbl 1160.11341. Page 346

Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
Tenebaum, Gérald (1995). Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics. Vol. 46. Translated by C.B. Thomas. Cambridge: Cambridge University Press. ISBN 0-521-41261-7. Zbl 0831.11001.

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Approximation by Riemann zeta zeros

References

See also