[hamdani yusuf (OP)]

Then compare to absolute value of points outside the critical line

When the real part gets higher than 1, the shape of the curve gets closer to sinusoidal.
plot |zeta (1+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%281%2Bsi%29%7C++from+0+to+30
plot |zeta (2+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%282%2Bsi%29%7C++from+0+to+30
plot |zeta (4+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%284%2Bsi%29%7C++from+0+to+30
plot |zeta (16+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%2816%2Bsi%29%7C++from+0+to+30

[hamdani yusuf (OP)]

When the real part gets lower than 0, the shape of the curve gets closer to exponential.
plot |zeta (0+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%280%2Bsi%29%7C++from+0+to+30
plot |zeta (-1+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%28-1%2Bsi%29%7C++from+0+to+30
plot |zeta (-5+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%28-5%2Bsi%29%7C++from+0+to+30

We can confirm these by plotting their logarithmic values.
plot ln|zeta (0+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+ln%7Czeta+%280%2Bsi%29%7C++from+0+to+30
plot ln|zeta (-1+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+ln%7Czeta+%28-1%2Bsi%29%7C++from+0+to+30
plot ln|zeta (-5+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+ln%7Czeta+%28-5%2Bsi%29%7C++from+0+to+30 
plot ln|zeta (-50+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+ln%7Czeta+%28-50%2Bsi%29%7C++from+0+to+30
plot ln|zeta (-500+si)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+ln%7Czeta+%28-500%2Bsi%29%7C++from+0+to+30
The last curve is close to straight line. It confirms that absolute value of zeta function gets closer to exponential when the real part gets lower than 0.

[hamdani yusuf (OP)]

These result in contradiction that there would be infinitely many zeros with the same imaginary parts, which brings us to conclude that Non-trivial zeros of Riemann's hypothesis must be located at the critical line, Re(s)=0.5, which means that Riemann's hypothesis must be true.

Now we're back to the first question in this thread.

What makes Riemann's Hypothesis Hard to Prove?

[Eternal Student]

Hi.

Now we're back to the first question in this thread.
......    What makes Riemann's Hypothesis Hard to Prove?

     The usual methods aren't productive.  That's the short answer.

   You've already had some reasonable replies from @Bored chemist  and @evan_au  (all on page 1) discussing the possibility that it might be true but unprovable from the ZF(C) axiom system.   Equally, it might be false but not provably false in the ZF(C) axiom system.   
    As you've probably already realised, it only takes someone to find a single counter-example:  A zero of the function which isn't an even integer or has real part = 1/2    and then the conjecture has been falsified.    Since people have been trying for a few years, the chances are now looking good that the conjecture is true.   The main issue is whether this is provable.

The bigger picture:
     We are used to being able to phrase or set up a problem in Mathematics and then find a proof (or proof that it doesn't hold).   Some human bias is then inevitable, you leave school thinking that mathematics can probably solve every problem.   Yes, sure there may be some problems I can't solve but someone, somewhere, could probably solve it or given enough time and determination I might even be able to solve it myself  - That's the sort of impression you leave school with.
     You would not have been shown or taught examples where Mathematics could not resolve a problem.   There would have been little educational value in it.   As a consequence, people rarely consider how many statements (let's say statements in formal first order logic) exist but have no proof (or falsification).   We have Godel's theorem's, so we know at least some such statements exist but few people consider how many such statements exist.
      Early Mathematicians were fairly certain that the rational numbers were all that was important and irrational numbers either didn't exist or were just inconvenient things that appear in mathematics only occasionally.   They were, in some sense, wrong.   The irrational numbers vastly outnumber the rational numbers (I know you've started other threads where the comparison of the cardinality of these sets has been discussed, so I'm not going to bother giving references for that).   Anyway, despite the seemingly overwhelming chances that numbers appearing at random would be irrational, it was possible for those early mathematicians to build incredibly powerful systems of mathematics, solve many problems that appeared in real life and improve the science and technology of the time.   They did all of that while skirting around what the majority of the Real numbers actually are.
        In a similar way, the number of mathematical statements*  that are NOT provably true or provably false from a given axiom system may be much larger than we had thought.

 * Mathematical statements  --->  statements constructed from a formal system of symbolic logic.   Reference:  https://en.wikipedia.org/wiki/Mathematical_logic#Formal_logical_systems ).

Best Wishes.

 

[hamdani yusuf (OP)]

The usual methods aren't productive.  That's the short answer.

Let's say I make a conjecture that there's no zero of Riemann's Zeta function with real component higher than 1. Is this conjecture provable?

[Eternal Student]

Hi.

Let's say I make a conjecture that there's no zero of Riemann's Zeta function with real component higher than 1. Is this conjecture provable?

   I don't know.   How long am I supposed to spend checking it?
Many people have spent some years looking at the Riemann Hypothesis,  I don't really want to spend that long.

Best Wishes.

Late Editing:  Actually this one may not take too long.   I think it's already been shown that all non-trivial zeros are in the critical strip   ( 0 <  Re(s) < 1).   So if   Re(s) > 1 then s is not a zero of the Riemann Zeta function.
Reference:  https://en.wikipedia.org/wiki/Riemann_zeta_function#Zeros,_the_critical_line,_and_the_Riemann_hypothesis

[hamdani yusuf (OP)]

Hi.

Let's say I make a conjecture that there's no zero of Riemann's Zeta function with real component higher than 1. Is this conjecture provable?

   I don't know.   How long am I supposed to spend checking it?
Many people have spent some years looking at the Riemann Hypothesis,  I don't really want to spend that long.

Best Wishes.

Late Editing:  Actually this one may not take too long.   I think it's already been shown that all non-trivial zeros are in the critical strip   ( 0 <  Re(s) < 1).   So if   Re(s) > 1 then s is not a zero of the Riemann Zeta function.
Reference:  https://en.wikipedia.org/wiki/Riemann_zeta_function#Zeros,_the_critical_line,_and_the_Riemann_hypothesis

Which part of the reference that you quoted contains the proof that zero of Riemann's zeta function can not have real part >1 ?

[hamdani yusuf (OP)]

It confirms that absolute value of zeta function gets closer to exponential when the real part gets lower than 0.

These results show that absolute value of zeta function in critical strip represents transition zone between exponential and sinusoidal function.

[hamdani yusuf (OP)]

My proof can be broken down into several parts to make it easier to analyze. It starts with an observation. To help tracing and pointing out in discussion, let's assign a number to it, #1.

The pattern we can observe is that for the same imaginary part, the plot's minima of zeta function gets closer to 0 when the real part of z is closer to 0.5

Part #2 is simply a consequence of part #1.

It means if there exist zero of zeta function where the real part of the variable is not equal to 0.5, the function will also produce zero when the real part of the variable is changed to get closer to 0.5

Part #3 is simply a consequence of part #2.

By reflection, if there exist zero of zeta function where the real part of the variable is not equal to 0.5, the function will also produce zero when the real part of the variable is changed to get further away from 0.5

Part #4 is additional information taken from external reference, which for now is taken as axiom.

But somehow it's no longer zero when Re(s)=1

Part#5 is the conclusion from previous parts, which is a proof by contradiction. This conclusion can be invalidated by showing that at least one of previous parts is false.

[hamdani yusuf (OP)]

At first glance, part#1 doesn't seem to be convincing, since it is only based on observations. But I'm pretty sure that it can be obtained from functional equation itself. So does the part#4 which the other parts are contradicted against.

[Eternal Student]

Hi.

Which part of the reference that you quoted contains the proof that zero of Riemann's zeta function can not have real part >1 ?

   The proof is not shown in that article,  just the final result is stated:


Riemann hypoth.JPG (80.23 kB . 1269x161 - viewed 2733 times)

   The same result is stated in may other articles,  for example:
 

Zeros of the Riemann zeta function zeta(s) come in two different types. So-called "trivial zeros" occur at all negative even integers s=-2, -4, -6, ..., and "nontrivial zeros" occur at certain values of t satisfying
s=σ+it    
for s in the "critical strip" 0<σ<1

[Taken from   https://mathworld.wolfram.com/RiemannZetaFunctionZeros.html ]

    I can't find the paper(s) where the result was first proved and it probably wasn't just one paper but several papers each with a result over different parts of the critical strip.   

This book seems to be cited by many articles and apparently does contain a full proof of the result and it's where I would recommend you start looking.   I'm sure that will also cite the original paper(s) where the result(s) were first proved, if you're intersted in the history.

E. C. Titchmarsh, The theory of the Riemann Zeta-function, Oxford Science publications, second edition, revised by D. R. Heath-Brown (1986).

     That's a £120 book and it's not on my book shelf but if you have access to a library you might be able to find it.

Best Wishes.

[hamdani yusuf (OP)]

The proof is not shown in that article,  just the final result is stated:

Without a proof, we are forced to appeal to the authority, which is a kind of logical fallacy.
Perhaps we can get the proof by scrutinizing the functional equation.

[Eternal Student]

Hi.

Perhaps we can get the proof by scrutinizing the functional equation.

   Maybe but that could take a long time.   The chances are it took several people a few years to piece that result together.   The fastest way to get the proof is likely to be getting to the library and finding the book.  The proof may not even be more than a few pages and it might even look quite easy - but that's the thing about Mathematics:   It often looks easy when you know how to do it.   However, getting there can involve multiple "dead-ends"  - time wasted in pursuing ideas that looked promising but ultimately lead nowhere useful.   Quite often there's a few steps where there is no good reason to explain why anyone would have thought that it was worth trying.

    If, like me, you can't easily get to a library, then you might prefer to make another search for the result on the internet.   Just be carefull to look for reliable and authoritative sources.  There are many supposedly complete proofs of the Riemann Hypothesis that are available online but most of these have been shown to be inaccurate.   These swamp the results you get in a simple Google search and just the process of searching for good information has already taken me nearly an hour.   Sadly, many of the reliable sources of information hold the articles and research papers behind a pay-wall  (you have to pay to view the article).

we are forced to appeal to the authority,

   No, you don't have to believe them because their "an authority".   However, it is obviously sensible and time efficient to check their proof first rather than trying to prove everything yourself from scratch.
   
Best Wishes.

[evan_au]

the thing about Mathematics:   It often looks easy when you know how to do it

I have heard that the mathematician Gauss had a reputation for working on a mathematical problem until he solved it. And then, knowing it was true, he looked for the most elegant way of showing that it was true - but in a way that made it difficult for other mathematicians to use the learnings he had gained in discovering the result.
https://mathworld.wolfram.com/NonconstructiveProof.html

[Eternal Student]

Hi.

I liked the post @evan_au .

I have heard that the mathematician Gauss had a reputation for working on a mathematical problem until he solved it.

   Yes, apparently Gauss gave up mathematics when he found he was too old and he couldn't go longer than about 6 hours without his mind starting to wander a bit.   (My mind wanders a bit after 10 minutes).

About a non-educational proof:
   The Riemann Hypothesis is very likely to be one of those things where we would learn more by seeing all the techniques that were tried but failed to prove the result.    If there is a final proof of the hypothesis, then it is likely to be very specific to this problem and may provide no transferable methods or techniques - i.e. nothing that would be especially useful to solve any other problem in mathematics.

    However, the link you provided to a definition of a "non-constructive proof" is really describing something else.   It isn't necessarily that there isn't any educational value in the proof.   Instead, it's just that the proof focuses on establishing just the existence of something without necessarily providing any method to actually find the object of interest.   Providing non-constructive proofs is a bit of a "fashion"  - but that's going to take too long to discuss in this post.  With computers there is now a renewed interest in providing more practical proofs in mathematics - things that will lead to algorithms for finding things.

Examples:
   The intermediate value theorem:  If f (x) is continuous on the interval [a, b] and Y is a value between f(a) and f(b), then there exists a number c ∈ (a, b) such that f(c) = Y .
   This is a common theorem and the proof does only what is required - it proves the existence of a suitable c∈(a,b).    However, it provides no practical method to actually evaluate c.   You might know that a suitable value must exist but your on your own if you have to find it.   The proof is "non-constructive" - it does not show any method you could use to construct (or find) this value, c, given a particular function f.

   The fundamental theorem of arithmetic:    Every Natural number, N, can be written as a product of primes raised to some power.   Furthermore that representation is unique upto changing the order in which you write the factors and perform the multiplication.
    This is another common theorem but the standard proof is much more practical.   It's a "constructive proof" or "construction proof" instead of just an existence proof.   The first part of the proof (establishing the existence of one representation as a product of prime factorisation) involves finding factors and progressively dividing them out.   If you were given a specific natural number like 10036 and had to find the unique prime factorisation, this is exactly how you would want to go about it.   So the proof does more than just show a unique factorisation exists - it shows you how to construct it.

Best Wishes.

[hamdani yusuf (OP)]

Here's an example of attempt to prove Riemann hypothesis.

https://www.math.purdue.edu/~branges/proof-riemann-2017-04.pdf

THE RIEMANN HYPOTHESIS
Louis de Branges*
Abstract. A proof of the Riemann hypothesis is to be obtained for the zeta functions
constructed from a discrete vector space of finite dimension over the skew–field of quaternions
with rational numbers as coordinates in hyperbolic analysis on locally compact Abelian groups
obtained by completion. Zeta functions are generated by a discrete group of symplectic
transformations. The coefficients of a zeta function are eigenfunctions of Hecke operators
defined by the group. In the nonsingular case the Riemann hypothesis is a consequence of
the maximal accretive property of a Radon transformation defined in Fourier analysis. In the
singular case the Riemann hypothesis is a consequence of the maximal accretive property of
the restriction of the Radon transformation to a subspace defined by parity. The Riemann
hypothesis for the Euler zeta function is a corollary.
1. Generalization of the Gamma Function

The abstract reminds me of some papers written by an old computer algorithm which constructed grammatically correct sentences but meaningless.

[hamdani yusuf (OP)]

Let's take s=a+bi, where a and b are real numbers.
0<a<0.5
0<b<∞

s*=a-bi
1-s=1-a-bi
1-s*=1-a+bi

when ζ(1-s)=0 → ζ(s)=0, ζ(s*)=0,  ζ(1-s*)=0
The sum and difference among those terms must also equal 0
ζ(s) + ζ(s*)=0
ζ(s) - ζ(s*)=0
ζ(1-s) + ζ(s*)=0
ζ(1-s) - ζ(s*)=0
ζ(1-s*) + ζ(s*)=0
ζ(1-s*) - ζ(s*)=0

ζ(s) + ζ(1-s)=0
ζ(s) - ζ(1-s)=0
ζ(1-s*) + ζ(1-s)=0
ζ(1-s*) - ζ(1-s)=0

ζ(s) + ζ(1-s*)=0
ζ(s) - ζ(1-s*)=0

[hamdani yusuf (OP)]

Let's take some examples of the plot, where a=0.3.
For some unclear reason, Wolfram Alpha refuses to plot |zeta (0.3+bi)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%280.3%2Bbi%29%7C++from+0+to+30
But it allows to plot |zeta (0.3+ti)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%280.3%2Bti%29%7C++from+0+to+30

ζ(s) + ζ(1-s*)  → ζ(0.3+ti) + ζ(0.7+ti)
plot |zeta (0.3+ti)+zeta (0.7+ti)|  from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%280.3%2Bti%29%2Bzeta+%280.7%2Bti%29%7C++from+0+to+30

ζ(s) - ζ(1-s*)  → ζ(0.3+ti) - ζ(0.7+ti)
plot |zeta (0.3+ti)- zeta (0.7+ti)| from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%280.3%2Bti%29-+zeta+%280.7%2Bti%29%7C+from+0+to+30

Plot of the sum looks more like ζ(0.5+ti), but if zoomed in, the minima are not 0.
Plot of the difference looks much different than ζ(0.5+ti). It doesn't even have the same base frequency.

[hamdani yusuf (OP)]

ζ(s) + ζ(1-s)  → ζ(0.3+ti) + ζ(0.7-ti)
plot |zeta (0.3+ti)+zeta (0.7-ti)| from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%280.3%2Bti%29%2Bzeta+%280.7-ti%29%7C+from+0+to+30

ζ(s) + ζ(1-s)  → ζ(0.3+ti) - ζ(0.7-ti)
plot |zeta (0.3+ti)-zeta (0.7-ti)| from 0 to 30
https://www.wolframalpha.com/input?i=plot+%7Czeta+%280.3%2Bti%29-zeta+%280.7-ti%29%7C+from+0+to+30

The plot of the sum looks more smooth near 0. While plot of the difference has twice the frequency.

[hamdani yusuf (OP)]

Basically, proofing Riemann's Zeta function must demonstrate that assuming the existence of non-trivial zero of Riemann's Zeta function with real component other than 0.5 leads to contradiction. Thousands of top mathematicians have tried to solve it unsuccessfully. Millions of math enthusiasts might have tried their luck attacking the problem from various directions, but it still withstands.

At this point, it should be obvious that direct attack on the problem is impossible. There must be some missing key ingredients not yet thought to be related to the problem.

Perhaps we can learn from someone else's success. It's possible that the problem here will end up with a similar story.
 

Old Problem About Mathematical Curves Falls to Young Couple
https://www.quantamagazine.org/old-problem-about-algebraic-curves-falls-to-young-mathematicians-20220825/
A basic fact of geometry, known for millennia, is that you can draw a line through any two points in the plane. Any more points, and you’re out of luck: It’s not likely that a single line will pass through all of them. But you can pass a circle through any three points, and a conic section (an ellipse, parabola or hyperbola) through any five.

More generally, mathematicians want to know when you can draw a curve through arbitrarily many points in arbitrarily many dimensions. It’s a fundamental question — known as the interpolation problem — about algebraic curves, one of the most central objects in mathematics. “This is really about just understanding what curves are,” said Ravi Vakil, a mathematician at Stanford University.



Larson began his involvement with the interpolation problem while he was working on another major question in algebraic geometry known as the maximal rank conjecture. When, as a graduate student, he set his sights on this conjecture — which had been open for more than a century — it seemed like “a really dumb idea, because this conjecture was like a graveyard,” Vakil said. “He was trying to chase something which people much older than him had failed at over a long period of time.”

But Larson kept at it, and in 2017, he presented a full proof that established him as a rising star in the field.

“They make the arguments seem very natural. Like, it seems very unsurprising,” said Dave Jensen, a mathematician at the University of Kentucky. “Which is odd, because this is a result that other people tried to prove and were unable to.”