The Riemann zeta function

[NTuft]

 

 

"There is a connection between physics and complex calculus, provided by harmonic functions in two-dimensional problems."

Can you explain that: harmonic functions, a la on the unit sphere, with real clear triangles?

@joigus,

Hello, good day,

What if the Riemann zeta function has primes distributed with (1/2) real part because of the way i is being used as a 'spanner' in the works?

As a placeholder it's made complex numbers a sort of math that can hold with limiting the nature of the imaginary component; or is it a component inseperable from an extra dimension like time, and so splits from the real number line for its expression? If complex math uses prime numbers under the square root operation in place as i', multiplication by the complex conjugate could still return results, and in particular the geometric morphisms of rotation on the complex plane, that are reducible like what is done currently with i as an operation placeholder, scaling the unit circle.

By Euler's equation we graph the triangular relations on a 1-length legged triangle in a sort of polar graphing for trigonemtric relations, on what's a unit circle bounded above and below by magnitudes of i from the approachable limit, 0, (0 being defined as a limit) or the asymptotic region; on a hyerbolic complex graph, with the hypberbola and conjugate hyperbola not centered on a unit circle's central (asymptotic) discontinuity (the eccentricity defining the circle): but, in its place an ellipsoid shape (ecc.= ?), bounded on the "real" axes by the 1/4 root of i' and with the upper and lower bounds imaginary parts 1/2 roots of i': the hyperbola gives us the logarithm's decay of exponentiation, and maps exponentiation by the euler number e base. So that map is extended to make hyperbolic angle or rapidity exponentiated with now harm to the triangles: but the triangles add to less than 180 on interior angles measured normally...

Again, in summary, the distribution of primes is best explained by explaining how complex math can use all such numbers as i, and the distribution of primes having (1/2) real part is a function of using +1 or -1 (which we will define as primes). Or, square root of -1, properly speaking: so we would have to map rules for exponentiations of i' now as a set of primes that is being alternated between values of +i', -i', +"1", -"1", by the operations of complex math. This looks like incorporating quaternion math or octonion math with the "+1s" and "-1s" let's call the approximation (i.e. the square root operation after carry-through) of the prime number: an irrational number we're not sure about, one which I think the set of which forms a relative uncountability between reals and complex numbers. 

Edited June 2, 2022 by NTuft
Introductary question re: complex calculus.

[joigus]

I don't understand what you're getting at. Take, eg., the sentence,

1 hour ago, NTuft said:

What if the Riemann zeta function has primes distributed with (1/2) real part because of the way i is being used as a 'spanner' in the works?

¿i being used as a spanner in the works? What does that mean?

The imaginary unit i does not come from a choice, as the vector (0,1) does, for example. Complex numbers are much more constrained than vectors, and obey different definitions. There is no such thing as rotational invariance for complex numbers, for example.

Quaternions are very different from complex numbers also. E.g., they're non-commutative. Serious maths are not about "this looks like that" and such.

So, while I don't understand what you mean, I see many problems with some of the things you say. Too loose connections in what I can understand from what you say.

Harmonic functions are the real and imaginary parts of differentiable functions of a complex variable when expressed in terms of x and y. They're used to represent 2-dimensional problems having to do with the Laplace equation (electrostatics, laminar fluids in absence of eddies, etc.) I don't know what that has to do with the Riemann hypothesis.

[NTuft]

10 minutes ago, joigus said:

¿i being used as a spanner in the works? What does that mean?

That the equations hold through using members of a set from i' as a value for i.

11 minutes ago, joigus said:

Quaternions are very different from complex numbers also. E.g., they're non-commutative. Serious maths are not about "this looks like that" and such.

[...]

12 minutes ago, joigus said:

I don't know what that has to do with the Riemann hypothesis.

As I understand it, the Riemann function can output the prime numbers by definition in a way, showing their distribution, with non-trivial values always 1/2 real part. The quaternion assignments, for example, as an extension of the complex numbers have a real component, and three complex components, all selected values of i=sqrt(-1), and it's non-commutative rules (as though they were time-ensconced components), or require taking vector quotients for space translations... Again, essentially that most all equations should hold.

Since you asked about complex calculus, I figured we'd take a function and see how it works through calculus with hyperbolic trigonometry and geometry.

[joigus]

2 hours ago, NTuft said:

Since you asked about complex calculus,

I did?

[NTuft]

17 hours ago, joigus said:

I did?

  In this way, 

On 8/15/2021 at 12:52 AM, joigus said:

I suppose what you wanna do here, @NTuft, is something like considering a complex variable [...] differentiating by it,
[...]

and then substituting, [...]

But be careful; you may be re-discovering complex calculus.

you at least broached the subject.

 

I am still hopeful you will expertly quash my amateur mathematical adventure. At this point I needed to be familiar with split complex numbers, with j² = +1 , and I was not familiar with that. You may not have to do anything. Thanks for taking some time to look at it, even if it is a nuisance.

 

P.s. : https://funkyenglish.com/idiom-throw-spanner-works/

Quote

Idiom – Throw a spanner in the works or Put a spanner in the works

Meaning – To do something that prevents an activity or plan from happening or being successful. This expression is used when something or someone introduces a problem, obstacle or something unexpected. A spanner in the works will cause delays to the progress of something in some way. This problem may be a result of carelessness – or sabotage.

In American English an adjustable spanner is called a monkey-wrench. As a result you might hear American people use the similar idiom – throw a monkey wrench in the works.

To remember this phrase imagine what would happen if somebody threw a spanner into a moving engine or machine. The spanner would likely cause the gears or machine parts to smash or stop working in some way.

In British English a spanner may also refer to a stupid or foolish person. “You forgot your wallet again? You are a complete spanner!” This usage is mildly offensive.

 

[joigus]

4 hours ago, NTuft said:

I am still hopeful you will expertly quash my amateur mathematical adventure. At this point I needed to be familiar with split complex numbers, with j² = +1 , and I was not familiar with that. You may not have to do anything. Thanks for taking some time to look at it, even if it is a nuisance.

 

Yes, you're right. It's me who brought it up. I forget what it had to do with the primes, though. Or how the question surfaced.

I'm no expert on number theory.

You may be on to something when you say that differentiable functions of a complex variable may have to do with the prime-counting problem. A function being differentiable in a complex variable z=x+iy is a much more restrictive condition than the corresponding condition on a real-variable function.

But in mathematics it's important to define anambiguously your concepts, and the state clearly what your assumptions are, and proving rigorously whatever you say about them.

Sorry, I can't contribute much more at this point. I hope that was helpful.

[NTuft]

On 6/2/2022 at 1:31 PM, NTuft said:

"There is a connection between physics and complex calculus, provided by harmonic functions in two-dimensional problems."

Can you explain that: harmonic functions, a la on the unit sphere, with real clear triangles?

 

On 6/2/2022 at 2:47 PM, joigus said:

Harmonic functions are the real and imaginary parts of differentiable functions of a complex variable when expressed in terms of x and y. They're used to represent 2-dimensional problems having to do with the Laplace equation [...]

The functions that satisfy the Laplace equation for spherical harmonics look a lot like our electron orbitals in Chemistry. I guess we have 3-Dimensional problems in chemistry, but not physics.

 

  

23 hours ago, joigus said:

But in mathematics it's important to define [un-, I gather]anambiguously your concepts, and the[n] state clearly what your assumptions are, and proving rigorously whatever you say about them.

Those 3 also necessary in logic, and your contribution is much appreciated. I did not see your reply until after replying. Thanks. The link to another line to follow up on is helpful.

Edited June 4, 2022 by NTuft
Accounting for prior reply.

[joigus]

13 hours ago, NTuft said:

The functions that satisfy the Laplace equation for spherical harmonics look a lot like our electron orbitals in Chemistry. I guess we have 3-Dimensional problems in chemistry, but not physics.

This is no coincidence. The Laplace equation must monitor the spatial/vacuum factor of any physical equation worth its salt, as Einstein observed in his famous popular book on the theory of relativity.

Physics does deal with many 3-dimensional problems, of course. When it comes to proposed generalisations of the standard model, it may even consider 26 dimensions.

Why do you say that? 

[NTuft]

9 hours ago, joigus said:

Why do you say that?

An unchecked quip from some negative emotion, as I'm quite attached to my pet theory. To continue "flying by the seat of my pants" I bandy about a posteriori findings as though I knew what I was talking about at the time.

 

Would you recommend any books on Physics? From @Genady I picked up mention of The Road to Reality by R. Penrose.

[joigus]

56 minutes ago, NTuft said:

Would you recommend any books on Physics? From @Genady I picked up mention of The Road to Reality by R. Penrose.

The Road to Reality is a very good book. Any recommendation from Genady is probably worth considering, OTOH.

It's not a book to actually learn physics though. It's more of a whirlwind tour of the exciting topics of modern physics.

As to popular books with emphasis on experiments, I recommend, Weinberg's The Discovery of Subatomic Particles.

Then you can try the Feynman Lectures on Physics.

If you want to learn physics in earnest, you probably can't do much better than Landau & Lifshitz's Course of Theoretical Physics. Encyclopedic (10 volumes.) A bit old, but will take you a long way in understanding the deepest principles of physics and how they're applied. There are many other books, I'm sure, and with a more modern focus. A good rule of thumb is: The more unassuming the title is, the more likely it will take you to the nuts and bolts.

I think you get the idea.

Although we could hardly be more off-topic.