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 2016-10-05, 03:57 #1 SteveC   Oct 2016 3 Posts Fourier Series for Prime Number Counting Functions If you're interested in prime number theory and the Riemann hypothesis, you might be interested in the following website where I've illustrated the genuine natural Fourier series for the base prime counting function Pi[x], Riemann's prime-power counting function, the first Chebyshev function, the second Chebyshev function, and a couple of additional related prime counting functions. http://www.primefourierseries.com/ The website illustrates fundamental relationships between the prime counting functions and their genuine natural Fourier series, such as the Fourier series for the first-order derivative of every prime counting function evaluates to 2f times the step size of the prime counting function at positive integer values of x, where f is the evaluation frequency limit and assumed to be a positive integer. This relationship holds for all evaluation frequencies including the minimum evaluation frequency f=1. As the evaluation frequency increases towards infinity, the primary lobe associated with a prime (or in some cases prime-power) becomes narrower and taller converging to the notion of a Dirac delta impulse function.
 2016-10-05, 04:27 #2 Dubslow Basketry That Evening!     "Bunslow the Bold" Jun 2011 40
 2016-10-05, 06:13 #3 Dubslow Basketry That Evening!     "Bunslow the Bold" Jun 2011 40
 2016-10-05, 17:06 #4 SteveC   Oct 2016 112 Posts Dubslow: I am working to document the general method for derivation of Fourier series for prime counting functions which I eventually plan to distribute, but at this point in time I'm only disclosing results of my research versus the underlying mathematics which are still under investigation for a possible proof of the Riemann Hypothesis. The first harmonic is primarily influenced by the first zeta zero. I'm not exactly sure whether you're interested in seeing an illustration of the influence of the second and third zeta zeros on the first harmonic of the Fourier series, or the influence of the second and third zeta zeros on the second and third harmonics of the Fourier series, but in both cases there doesn't seem to be an obvious influence such as the influence illustrated of the first zeta zero on the first harmonic.
2016-10-07, 16:45   #5
xilman
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Quote:
 Originally Posted by SteveC If you're interested in prime number theory and the Riemann hypothesis, you might be interested in the following website where I've illustrated the genuine natural Fourier series for the base prime counting function Pi[x], Riemann's prime-power counting function, the first Chebyshev function, the second Chebyshev function, and a couple of additional related prime counting functions. http://www.primefourierseries.com/
This awakens a vague recollection about testing Goldbach's conjecture through Fourier analysis. I read about it some years ago but I'm no longer sure where. C&P perhaps?

2016-10-07, 16:49   #6
CRGreathouse

Aug 2006

598710 Posts

Quote:
 Originally Posted by xilman C&P perhaps?

2016-10-07, 16:54   #7
science_man_88

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Jul 2009
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203428 Posts

Quote:
 Originally Posted by xilman This awakens a vague recollection about testing Goldbach's conjecture through Fourier analysis. I read about it some years ago but I'm no longer sure where. C&P perhaps?

2016-10-07, 17:25   #8
xilman
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Quote:
 Originally Posted by CRGreathouse I don't remember reading about it there, FWIW.
I'll check up when I get home. On vacation in Spain right now.

2016-10-07, 18:05   #9
CRGreathouse

Aug 2006

5,987 Posts

Quote:
 Originally Posted by xilman I'll check up when I get home. On vacation in Spain right now.
Have fun!

2016-10-09, 11:29   #10
xilman
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Quote:
 Originally Posted by CRGreathouse Have fun!
Yup. C&P. It's in exercise 9.79 in both 1st and 2nd editions.

This question is motivated by the text in pp 491--2 in the second edition and pp 446--7 in the first.

 2016-10-14, 21:48 #11 SteveC   Oct 2016 310 Posts Evolution of Zeta Zeros from Second Chebyshev Function I have now illustrated the evolution of the zeta zeros from the Mellin transform of the Fourier series for the first-order derivative of the second Chebyshev function. http://www.primefourierseries.com/?page_id=961

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