Playing with Willans Formulae and Analytic Step Function Approximations (and actual step functions)

[Jarhyn]

Jarhyn,

As long as you are enjoying yourself I think hat is all that really matters.


As to triangle waves and Fourier series.

Put a sine wave into an audio amp and you hear a single tone. Combine two sines and you hear a sound. Put the combined signal through a filter that rejects one of the sines and you hear only a single tone.

A square or triangle wave has a series that describes the single sines that make up the signal. Put a triangle wave trough a bandpass filter that selects one of the sines in its Fourier series and you hear the frequency.

Put a square wave with a frequency of 1kz through a low pass filter that cuts off above 1kz and you get a 1kz sine wave.

That's the thing though... The triangle wave is the arcsin of the sin.

Anyway, I discussed the whole thing with a coworker who has his master's in math, and physics too, I think?

He managed to follow along with it as I put it all together for him, and we came out of it both agreeing that finding the fourier series of tanh sums is a worthwhile endeavor because they're "smooth" in a way that the absolute value version is not, and smoothness is one of my goals as to how to improve the result.

 

[steve_bank]

If you have a signal generator and a spectrum or signal analyzer at work you can put square and triangle waves into the analyzer and see the spectrum.

One experiment is worth a thousand words.

If not you can put the signals you generate in code into the Numpy FFT and see the spectrum.

 

[Jarhyn]

If you have a signal generator and a spectrum or signal analyzer at work you can put square and triangle waves into the analyzer and see the spectrum.

One experiment is worth a thousand words.

If not you can put the signals you generate in code into the Numpy FFT and see the spectrum.

I'm not sure any of our signal generators can generate a pattern of tanh sums, though

Honestly, it would be more straightforward to just synthesize waves with alternating sums of tanh and see what falls out of a FFT, like you said.

 

[Jarhyn]

I am actually curious what you would even call the arcsin(sin(x)) operation.

Yes it's a triangle wave but it's also an operation on numbers. Reflectulo, after modulo, as it creates a reflection of a beam between two planes, and as the triangle is to reflectulo, the sawtooth is to modulo?

 

[steve_bank]

Arbitrary waveform generators can synthesize any waveform.

Python has sound processing functions.

There is audio oscilloscope and, waveform generator, and spectrum analyzer software They take data froand output to the sound input and outputs.

I had it wrong. arcsin(sin()) does not equal sin(arcsin()).

 

[Jarhyn]

Arbitrary waveform generators can synthesize any waveform.

Python has sound processing functions.

There is audio oscilloscope and, waveform generator, and spectrum analyzer software They take data froand output to the sound input and outputs.

I had it wrong. arcsin(sin()) does not equal sin(arcsin()).

Yeah, arcsin has a tightly bounded defined range, and part of the reason for feeding a sin into it was that sin produces bounded values valid in arcsin.

I'll take a look at it later when I'm off work, granted my computer is shit.

After consulting with some coworkers, it does look like this may actually be a solution for the Riemann hypothesis but I'm taking it slow this time and I am going to dot all my I's and cross all my T's before publishing or making any firm claims?

Anyway, I have another coworker besides the one I explained the proof of function to, and they're gonna ask around to see if this satisfies the magic that is expected of such a function to declare it "a solution" to the "hard problem" that is Riemann.

Looking around at other posts on Reddit, it does look like the J function solves an unsolved problem, though, so it's got merit for publishing either way, even if it's not "The Holy Grail".

As an aside, Steve, would you like some credit on this? I can credit you in the proof as a co-author, if this is a discovery.

 

[steve_bank]

Thanks, but I have not really contributed to your thesis, it is your baby.

I was never interested in research and publishing, I just liked building thing and solving problems. A worker bee engineer.

 

[Swammerdami]

So, it should be enough that it makes me happy, but how happy it can make me depends on part on how much I actually accomplish as objective progress.

I appreciate your efforts, Jarhyn, and am sorry I've been of no help. I spend much time on a variety of things^* but do VERY little math or even programming these days. (Frittering away my time is the story of my life )
* - For example I spent several hours writing a "Show-and-Tell" about the Y-chromosome tree, and that was just synopsizing my readings. I find the Y-tree story fascinating, but am afraid few others do.

Regarding prime numbers, have you read Prime Obsession by John Derbyshire? (Especially Chapter 7 "The Golden Key".) There MIGHT be a relationship between the "Golden Key" and what you're trying to do.

 

[Jarhyn]

So, it should be enough that it makes me happy, but how happy it can make me depends on part on how much I actually accomplish as objective progress.

I appreciate your efforts, Jarhyn, and am sorry I've been of no help. I spend much time on a variety of things^* but do VERY little math or even programming these days. (Frittering away my time is the story of my life )
* - For example I spent several hours writing a "Show-and-Tell" about the Y-chromosome tree, and that was just synopsizing my readings. I find the Y-tree story fascinating, but am afraid few others do.

Regarding prime numbers, have you read Prime Obsession by John Derbyshire? (Especially Chapter 7 "The Golden Key".) There MIGHT be a relationship between the "Golden Key" and what you're trying to do.

Ok, so, I spent most of my afternoon reading that book. I got to chapter 15 skipping most of the history. Chapter 15 is what I was really digging for, and the complex plane stuff... Well, it actually makes more sense now when looking at the problem through a lens of Fourier analysis and seeing that I'm probably simulating an imposition of absolute value on a thing, forcing an imposition to rotate only at precisely controlled times in the system relating to shifts of phase in a system.

Anyway, chapter 15 discussed the major point about the relationships between O sqrt(x) ln being the computational bound that proves Riemann, and that's what I achieved. I'm just not sure if it actually gives a pathway to discovering the error term on Li, because it skipped right past that to indication.

I suggest also reading chapter 15 again if you haven't. I'm going to keep reading more this week.

Last time has me so worried, more than ever, I'm going to say I did more than I actually accomplished and embarrass myself again!

 

[steve_bank]

Cretatinga square wave with sine waves using a Fourier Series.

A square wave is comprised on the fund metal frequency plus the odd shamanic multiplied by a coefficient.

Plenty of information on the net. You can find coefficients and the harmonics for a triangle wave.

Run the code for nterms = 10,100, and 1000.

import matplotlib.pyplot as plt
import array as ar
import math as ma

nsignal = 1000
nterms = 100
y = ar.array('d',nsignal*[0])
ys = ar.array('d',nsignal*[0])
t = ar.array('d',nsignal*[0])

dt = 1/(nsignal-1)
for i in range(nsignal):t[i] = i *dt  # 0 to 1 second

#square wave fundmntal plus odd harmonics

f = 1  # fundamntal 1 hertz
for j in range(nterms):
    coef = 4/(f*ma.pi)
    for i in range(nsignal):y[i] = coef * ma.sin(2*ma.pi*f*t[i])
    for k in range(nsignal):ys[k] += y[k]  # sum the harmonics
    f += 2  # next odd harmonic

plt.plot(t,ys,linewidth=2.0,color="k")
plt.grid(color = 'k',linestyle= '--',linewidth = 1)
plt.show()

 

[steve_bank]

Cretatinga square wave with sine waves using a Fourier Series.

A square wave is comprised on the fund metal frequency plus the odd shamanic multiplied by a coefficient.

Plenty of information on the net. You can find coefficients and the harmonics for a triangle wave.

Run the code for nterms = 10,100, and 1000.

import matplotlib.pyplot as plt
import array as ar
import math as ma

nsignal = 1000
nterms = 100
y = ar.array('d',nsignal*[0])
ys = ar.array('d',nsignal*[0])
t = ar.array('d',nsignal*[0])

dt = 1/(nsignal-1)
for i in range(nsignal):t[i] = i *dt  # 0 to 1 second

#square wave fundmntal plus odd harmonics

f = 1  # fundamntal 1 hertz
for j in range(nterms):
    coef = 4/(f*ma.pi)
    for i in range(nsignal):y[i] = coef * ma.sin(2*ma.pi*f*t[i])
    for k in range(nsignal):ys[k] += y[k]  # sum the harmonics
    f += 2  # next odd harmonic

plt.plot(t,ys,linewidth=2.0,color="k")
plt.grid(color = 'k',linestyle= '--',linewidth = 1)
plt.show()

Change dt = 1/(nsignal-1) to dt = 2/(nsignal-1), and f += 2 to f += 1 and you will see a cyclic ramp.