Power spectrum of the fluctuation of Chebyshev's prime counting function

• Published in: Physica A Vol. 361; no. 1; pp. 35 - 40
• Main Authors: Lan, Boon Leong, Yong, Shaohen
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15-02-2006
• Subjects: Chebyshev's prime counting function; Fluctuation; Power-law power spectrum; Riemann hypothesis; Time series; Riemann hypothesis; Power-law power spectrum; Fluctuation; Time series; Chebyshev's prime counting function
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2005.06.080

The one-sided power spectrum of the fluctuation of Chebyshev's weighted prime counting function is numerically estimated based on samples of the fluctuating function of different sizes. The power spectrum is also estimated analytically for large frequency based on Riemann hypothesis and the exact formula for the fluctuating function in terms of all the non-trivial Riemann zeroes. Our analytical estimate is consistent with our numerical estimate of a
1
/
f
2
power spectrum.