Fourier Series for Prime Number Counting Functions

SteveC · 2016-10-05, 03:57

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.

Dubslow · 2016-10-05, 04:27

Shouldn't you write this up and submit it to a journal? Or at least some sort of paper posted somewhere.

Dubslow · 2016-10-05, 06:13

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

SteveC · 2016-10-05, 17:06

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.

xilman · 2016-10-07, 16:45

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?

CRGreathouse · 2016-10-07, 16:49

Quote:

Originally Posted by xilman

C&P perhaps?

I don't remember reading about it there, FWIW.

science_man_88 · 2016-10-07, 16:54

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?

https://www.google.ca/webhp?sourceid...urier+analysis suggest any of 16,500 google results may help.

xilman · 2016-10-07, 17:25

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.

CRGreathouse · 2016-10-07, 18:05

Quote:

Originally Posted by xilman

I'll check up when I get home. On vacation in Spain right now.

Have fun!

xilman · 2016-10-09, 11:29

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.

SteveC · 2016-10-14, 21:48

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

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, 06:13	#3
Dubslow Basketry That Evening! "Bunslow the Bold" Jun 2011 40<A<43 -89<O<-88 3·29·83 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 2016-10-05 at 06:16

2016-10-14, 21:48	#11
SteveC 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 first-order derivative of the second Chebyshev function. http://www.primefourierseries.com/?page_id=961

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2016-10-05, 04:27	#2
Dubslow 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.

2016-10-05, 17:06	#4
SteveC Oct 2016 3₁₀ 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.