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Old 2016-10-05, 03:57   #1
SteveC
 
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Default 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.
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Old 2016-10-05, 04:27   #2
Dubslow
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Shouldn't you write this up and submit it to a journal? Or at least some sort of paper posted somewhere.
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Old 2016-10-05, 06:13   #3
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Er, I take that back, seems to be pretty much just a bunch of plots. I had thought there was some actual research.

Very nice graphs to be fair. Section 9 could I think benefit from graphs showing the second and third zeros as well.

Also the navigation is a bit obtuse

Last fiddled with by Dubslow on 2016-10-05 at 06:16
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Old 2016-10-05, 17:06   #4
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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.
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Old 2016-10-07, 16:45   #5
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Quote:
Originally Posted by SteveC View Post
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?
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Old 2016-10-07, 16:49   #6
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C&P perhaps?
I don't remember reading about it there, FWIW.
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Old 2016-10-07, 16:54   #7
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Quote:
Originally Posted by xilman View Post
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?
https://www.google.ca/webhp?sourceid...urier+analysis suggest any of 16,500 google results may help.
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Old 2016-10-07, 17:25   #8
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I don't remember reading about it there, FWIW.
I'll check up when I get home. On vacation in Spain right now.
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Old 2016-10-07, 18:05   #9
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I'll check up when I get home. On vacation in Spain right now.
Have fun!
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Old 2016-10-09, 11:29   #10
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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.
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Old 2016-10-14, 21:48   #11
SteveC
 
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Default 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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