20161005, 03:57  #1 
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 primepower 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 firstorder 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 primepower) becomes narrower and taller converging to the notion of a Dirac delta impulse function. 
20161005, 04:27  #2 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
Shouldn't you write this up and submit it to a journal? Or at least some sort of paper posted somewhere.

20161005, 06:13  #3 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
1C35_{16} Posts 
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 20161005 at 06:16 
20161005, 17:06  #4 
Oct 2016
3 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. 
20161007, 16:45  #5  
Bamboozled!
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May 2003
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Quote:


20161007, 16:49  #6 
Aug 2006
3×1,993 Posts 

20161007, 16:54  #7  
"Forget I exist"
Jul 2009
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Quote:


20161007, 17:25  #8 
Bamboozled!
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20161007, 18:05  #9 
Aug 2006
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20161009, 11:29  #10 
Bamboozled!
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20161014, 21:48  #11 
Oct 2016
3 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 firstorder derivative of the second Chebyshev function.
http://www.primefourierseries.com/?page_id=961 
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