https://en.wikipedia.org/wiki/Riemann_hypothesis
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part
1/2
. Many consider it to be the most important unsolved problem in pure mathematics.[1] It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.
The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.
The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:
The real part of every nontrivial zero of the Riemann zeta function is
1/2
.
Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers
1/2 + i t, where t is a real number and i is the imaginary unit.
Visualizing the Riemann zeta function and analytic continuation
A Youtube channel seems to be dedicated to explain this problem, and I find it as one of the best explanation online.If you are serious to understand this problem, this video is a good place to start.
https://www.youtube.com/c/zetamath/videos
Factorials, prime numbers, and the Riemann Hypothesis
Today we introduce some of the ideas of analytic number theory, and employ them to help us understand the size of n!. We use that understanding to discover a surprisingly accurate picture of the distribution of the prime numbers, and explore how this fits into the broader context of one of the most important unsolved problems in mathematics, the Riemann Hypothesis.
The Basel Problem Part 1: Euler-Maclaurin Approximation
This is the first video in a two part series explaining how Euler discovered that the sum of the reciprocals of the square numbers is π^2/6, leading him to define the zeta function, and how Riemann discovered the surprising connection between the zeroes of the zeta function and the distribution of the primes, leading ultimately to his statement of the Riemann Hypothesis. This video focuses on how Euler developed a method to approximate this sum to 17 decimal places, as well as how the Bernoulli numbers naturally appear as part of this problem.
The Basel Problem Part 2: Euler's Proof and the Riemann Hypothesis
In this video, I present Euler's proof that the solution to the Basel problem is pi^2/6. I discuss a surprising connection Euler discovered between a generalization of the Basel problem and the Bernoulli numbers, as well as his invention of the zeta function. I explain Euler's discovery of the connection between the zeta function and the prime numbers, and I discuss how Riemann's continuation of Euler's work led him to state the Riemann hypothesis, one of the most important conjectures in the entire history of mathematics.
Sections of this video:
00:00 Intro
01:24 Euler's Basel proof
23:20 The zeta function and the Bernoulli numbers
32:01 Zeta and the primes
48:15 The Riemann hypothesis
Analytic Continuation and the Zeta Function
Where do complex functions come from? In this video we explore the idea of analytic continuation, a powerful technique which allows us to extend functions such as sin(x) from the real numbers into the complex plane. Using analytic continuation we can finally define the zeta function for complex inputs and make sense of what it is the Riemann Hypothesis is claiming.
Chapters:
00:00 zetamath does puzzles
00:23 Recap
02:40 Bombelli and the cubic formula
08:45 Evaluating real functions at complex numbers
12:33 Maclaurin series
21:22 Taylor series
27:19 Analytic continuation
35:57 What goes wrong
48:19 Next time
Complex Integration and Finding Zeros of the Zeta Function
In this video we examine the other half of complex calculus: integration. We explain how the idea of a complex line integral arises naturally from real definite integrals via Riemann sums, and we examine some of the properties of this new sort of integral. In particular, we consider some complications that arise when trying to apply the fundamental theorem of calculus to complex functions.
We then bring these ideas to the central focus of this series: the zeta function and the Riemann hypothesis. By the end of the video, we will be able to use complex integrals to approximate the location of the zeroes of the zeta function (or those of any other complex function for that matter)!
https://en.wikipedia.org/wiki/Riemann_zeta_function#Riemann's_functional_equation
This zeta function satisfies the functional equation
(https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c33a4fef2cb6271b95028cf7a012d2c0d0ec00)
where Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product sin(πs/2)Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor.
https://en.wikipedia.org/wiki/Riemann_zeta_function#Other_results
The fact that
(https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b09034e8f5f6df4406385816e4a6f7109e19cc)
for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = ½
It is also known that no zeros lie on a line with real part 1.
https://en.wikipedia.org/wiki/Riemann_hypothesis#Zeros_on_the_critical_line
Hardy (1914) and Hardy & Littlewood (1921) showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function. Selberg (1942) proved that at least a (small) positive proportion of zeros lie on the line. Levinson (1974) improved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and Conrey (1989) improved this further to two-fifths. In 2020, this estimate was extended to five-twelfths by Pratt, Robles, Zaharescu and Zeindler[22] by considering extended mollifiers that can accommodate higher order derivatives of the zeta function and their associated Kloosterman sums.
https://mathworld.wolfram.com/RiemannHypothesis.html
It is known that the zeros are symmetrically placed about the line I(s)=0. This follows from the fact that, for all complex numbers s,
1. s and the complex conjugate s* are symmetrically placed about this line.
2. From the definition (1), the Riemann zeta function satisfies zeta(s*)=zeta(s)*, so that if s is a zero, so is s*, since then zeta(s*)=zeta(s)*=0*=0.
It is also known that the nontrivial zeros are symmetrically placed about the critical line R(s)=1/2, a result which follows from the functional equation and the symmetry about the line I(s)=0. For if s is a nontrivial zero, then 1-s is also a zero (by the functional equation), and then 1-s* is another zero. But s and 1-s* are symmetrically placed about the line R(s)=1/2, since 1-(x+iy)*=(1-x)+iy, and if x=1/2+x', then 1-x=1/2-x'.
There must be some missing key ingredients not yet thought to be related to the problem.No
Why? Do you have any reason to think that way? Or is it just your intuition?There must be some missing key ingredients not yet thought to be related to the problem.No
It may not be provable.
It is known that some problems are impossible to solve- That's the incompleteness theorem.Why? Do you have any reason to think that way? Or is it just your intuition?There must be some missing key ingredients not yet thought to be related to the problem.No
It may not be provable.
There must be some missing key ingredients not yet thought to be related to the problem.is false.
some problems are impossible to solve- That's the incompleteness theoremThe Incompleteness Theorem applies within a specific domain of mathematics - it may be impossible to prove some true statements within the axioms of that system.
At this point, it should be obvious that direct attack on the problem is impossible.
Here are some interesting results which might be useful in solving the problem.When Riemann zeta function produces 0 result, ζ(s) = 0, at least one of these terms is 0Quotehttps://en.wikipedia.org/wiki/Riemann_zeta_function#Riemann's_functional_equation
This zeta function satisfies the functional equation
(https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c33a4fef2cb6271b95028cf7a012d2c0d0ec00)
where Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product sin(πs/2)Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor.
There may be additional true statements within the new, extended set of axioms that are unproveable within that extended set of axioms.And the RH might be one of them.
From above results, it can be inferred that for any point in the critical strip,
ζ(s*)=ζ(1-s) if and only if s is in critical line,
where s*=complex conjugate of s.
4) ζ(1-s) = 0 = ζ(s)
The fact thatBut we only care about the case where ζ(s) = 0.
ζ(s) = (ζ(s*))*
Let's learn what history can teach us.There may be additional true statements within the new, extended set of axioms that are unproveable within that extended set of axioms.And the RH might be one of them.
Is there proof that there is a proof, or is it possible that the OP is wasting his time?
When Riemann zeta function produces 0 result, ζ(s) = 0, at least one of these terms is 0Riemann's hypothesis only cares about point #4, which gives non-trivial zeros.
1) 2s = 0 → s = -∞
2) πs-1 = 0 → s = -∞
3) sin(πs/2).Γ(1-s) = 0 → s ∈ {negative even numbers}
4) ζ(1-s) = 0 = ζ(s)
Point #3 gives trivial zeros, while point #4 gives non-trivial zeros.
when 1-s=s → 1=2s → s=½
But ζ(½) <> 0
https://en.wikipedia.org/wiki/Riemann_zeta_function#Other_resultsUsing above results, proving Riemann's hypothesis can be done by following these reasonings
The fact that
(https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b09034e8f5f6df4406385816e4a6f7109e19cc)
Or if written in alternative notation, ζ(s)=(ζ(s*))*
for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = ½
It is also known that no zeros lie on a line with real part 1.
Then compare to absolute value of points outside the critical lineWhen the real part gets higher than 1, the shape of the curve gets closer to sinusoidal.
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.
What makes Riemann's Hypothesis Hard to Prove?
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.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?
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?
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 ?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
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.
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
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
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
But somehow it's no longer zero when Re(s)=1
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:
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
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.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.
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.
the thing about Mathematics: It often looks easy when you know how to do itI 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.
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).
THE RIEMANN HYPOTHESISThe abstract reminds me of some papers written by an old computer algorithm which constructed grammatically correct sentences but meaningless.
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
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.
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.
(https://d2r55xnwy6nx47.cloudfront.net/uploads/2022/08/BRILL_NOETHER_CURVES_920-desktop.svg)
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.”
There is actually a formula for the nth prime number! One was cleverly engineered in 1964 by C. P. Willans. But is it useful?
# Python code
import math
def prime(n):
return 1 + sum([
math.floor(pow(n/sum([
math.floor(pow(math.cos(math.pi * (math.factorial(j - 1) + 1)/j), 2))
for j in range(1, i+1)
]), 1/n))
for i in range(1, pow(2, n)+1)
])
--------------
(* Mathematica code *)
prime[n_] := 1 + Sum[Floor[(n/Sum[Floor[Cos[Pi ((j - 1)! + 1)/j]^2], {j, 1, i}])^(1/n)], {i, 1, 2^n}]
--------------
0:00 A formula for primes?
1:24 Engineering a prime detector
4:00 Improving the prime detector
5:46 Counting primes
6:29 Determining the nth prime
9:42 The final step
11:36 What counts as a formula?
12:56 What's the point?
13:51 Who was Willans?
Yitang Zhang, a Chinese-American mathematician, reportedly disclosed in an online salon organized by the Peking University Alumni Association on October 15 that he has proven the longstanding Landau-Siegel zeros theory. This finding is related to the Riemann hypothesis, a formula for the distribution of prime numbers that has remained unsolved for more than a century. However, the claim has not yet been fully verified, and it is reported that a relevant article of more than 100 pages will be sent to a preprint website in early November.
The Landau-Siegel zeros topic has represented one of the most difficult problems in number theory this century. It is a weak form of the Riemann hypothesis, which studies the existence of zeros in the DirichletL-function (a function defined on the whole complex plane). A century of research has shown that the Landau-Siegel zeros can be more difficult to solve than the Riemann hypothesis. Therefore, if Zhang Yitang has really proven that Landau-Siegel zeros exist, the Riemann hypothesis would be wrong. But for now, many people are more inclined to believe that Zhang proved the opposite result.
Solve one equation and earn a million dollars! We will explorer the secrets behind the Riemann Hypothesis - the most famous open problem in mathematics - and what it would tell us about prime numbers.
I should have mentioned one additional property, namely zeros are mirrored along the line 1/2, even though non of them are found and maybe even non of them even exist. This way, every zero not on the line would give a harmonic with Re(s) greater than 1/2, thereby breaking the estimates for the prime counting function.
This video proves the Riemann hypothesis. Otherwise, you get a $10,000 reward. (M1)Just in case you have time to spare.
In this short (4:33) video, you will see the proof of the Riemann Hypothesis. I will give you $10,000 If you provide a numeric counterexample showing the video fails to prove the Riemann Hypothesis. Also, please keep in mind that you can share this video to win the $10,000 cash reward. Please post your counterexample in the comments.
Direct messaging and email info link below
https://www.0bq.com/contact
Note that If your counterexample disproves Riemann's hypothesis, it will be your responsibility to document it before sharing it with anybody. I will try my best to protect your right, and I will not claim any ownership of your counterexample, and you will win the $10,000 cash reward regardless of any other prizes.
RIEMANN HYPOTHESIS PROOF / SOLVED, 17584 VIEWS, 0 COUNTEREXAMPLE! (M2)
Over the years, hundreds of mathematical theories have been built upon the assumption that Riemann's last theorem is true. Therefore, considerable efforts have been made by several of the best mathematical minds around the world to protect the legitimacy of these theories. However, in this video we have finally proved this famous theorem that had resisted all efforts to be proven for over one and a half centuries.
Step Zero of Analytic Continuation Gateway to the Riemann Hypothesis
Like origami art, where we can fold a paper to create a boat and then refold the same paper differently to build a totally different thing (for example, a bird), this simple video shows that we can unfold an infinite-divergence series (commonly known as Riemann's zeta function) and then refold it to get a finite value. It is fascinating to see unfolding a divergence function in 6 steps and then refolding the same function in 6 steps gives us a convergent function. This is the most elegant method to analytically continue Riemann's zeta function to the critical strip because it shows mathematics overlaps with art.
What is this RH all about?
In Simplest of terms...
Is it finding a Pattern or Function that could precisely Predict the list of Prime Numbers?
This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line s = 1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1. He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.
https://en.wikipedia.org/wiki/Riemann_hypothesis#Origin
https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences
The practical uses of the Riemann hypothesis include many propositions known to be true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis.
(Pls don't say they will search online & copy/paste & give a printout within minutes)It's likely what they will do.
(Pls don't say they will search online & copy/paste & give a printout within minutes)It's likely what they will do.
Alternatively, they will just tell you to ask ChatGPT.
Have you watched this video?(Pls don't say they will search online & copy/paste & give a printout within minutes)It's likely what they will do.
Alternatively, they will just tell you to ask ChatGPT.
You know what i was trying to Understand by asking that question was, is there an equation or function for getting Prime Numbers.
I'm supposing one can simply use the Calculator.
Or a simplified program in the Computer.
(by ways of checking each number like 1234567 etc & reaching the answer)
But...is there a Pattern to predict Primes without Calculations?
I just found a great video about prime numbers I'd like to share here.
A Formula for the Primes: Willans' FormulaQuoteThere is actually a formula for the nth prime number! One was cleverly engineered in 1964 by C. P. Willans. But is it useful?
# Python code
import math
def prime(n):
return 1 + sum([
math.floor(pow(n/sum([
math.floor(pow(math.cos(math.pi * (math.factorial(j - 1) + 1)/j), 2))
for j in range(1, i+1)
]), 1/n))
for i in range(1, pow(2, n)+1)
])
--------------
(* Mathematica code *)
prime[n_] := 1 + Sum[Floor[(n/Sum[Floor[Cos[Pi ((j - 1)! + 1)/j]^2], {j, 1, i}])^(1/n)], {i, 1, 2^n}]
--------------
0:00 A formula for primes?
1:24 Engineering a prime detector
4:00 Improving the prime detector
5:46 Counting primes
6:29 Determining the nth prime
9:42 The final step
11:36 What counts as a formula?
12:56 What's the point?
13:51 Who was Willans?
Has anyone noticed a peculiar thing around zeta(-41)?This is not the only jump found in zeta function. You can find the other numbers where the argument of zeta function switches value around negative odd integers. So far, the distribution of the jump looks random.
The harmonic numbers are the partial sums of the harmonic series - sums of whole number reciprocals. This video explores how we can extend their domain to the entire real line.
This is my entry for the Summer of Math Exposition 1.
00:00 - Intro
1:45 - Graphing the Harmonic Numbers
2:47 - A Recursive Formula
4:23 - Using the Recursive Formula
7:33 - The Super Recursive Formula
8:52 - Finding the Interval
11:27 - Example: H(0.5)
11:59 - Deriving the Solution
13:10 - Graphing the Solution
In this video, I walk through the derivation of an extension of the factorial function that works for any number: fractional, irrational, and even complex! This turns out to be a very important function, known as the gamma function, which has many surprising connections, one of which I explore in the last chapter of the video.
Chapters:
0:00 Introduction
1:38 A few Disclaimers
3:58 The Recursive Formula
6:50 The Super Recursive Formula
8:45 A minor setback
10:28 Logarithms
15:21 Deriving the Solution
19:26 Our Constraints
20:25 History and Conventions
22:16 The Miracle
25:44 The End
In this video, we build on my last two videos by exploring connections between the gamma function (the extended factorials), the digamma function (the extended harmonic numbers), and trigonometry. We derive Euler's Sine Product Formula, which we then use to prove the gamma and digamma functions' reflection formulas. Finally, we derive a related formula for calculating cotangent.
Chapters:
00:00 Intro
0:43 Background and Notation
3:24 The Digamma-Cotangent Connection
5:09 The Gamma-Sine Connection
6:04 The Sine Product Formula
9:59 Proving the Gamma-Sine Connection
12:22 The value of (1/2)!
13:07 Proving the Digamma-Cotangent Connection
14:21 The True Logarithmic Derivative
15:52 An Infinite Sum for Cotangent
17:46 Final Thoughts
When Riemann zeta function produces 0 result, ζ(s) = 0, at least one of these terms is 0Since ζ(s)* = ζ(s*)
1) 2s = 0 → s = -∞
2) πs-1 = 0 → s = -∞
3) sin(πs/2).Γ(1-s) = 0 → s ∈ {negative even numbers}
4) ζ(1-s) = 0 = ζ(s)
Point #3 gives trivial zeros, while point #4 gives non-trivial zeros.
when 1-s=s → 1=2s → s=0.5
But ζ(0.5) <> 0
ζ(s) + ζ(1-s*)=0Let's plot those equations with values of s where the real part is 0.5.
ζ(s) - ζ(1-s*)=0
What if you could group the trivial zeros of the Riemann zeta function with any hypothetical non-trivial zero off the critical line. And what if this group could not exist within the critical strip? Wouldn't this proof that the Riemann hypothesis is correct?
But...is there a Pattern to predict Primes without Calculations?
When a small imaginary number is added to the plot, we get different peculiarity.I'm now pretty sure that the jiggy curve is caused by truncated calculations of the numeric method.
https://www.wolframalpha.com/input?i=plot%28im%28zeta+%28x%2Bi%2F10e13%29%29%29+from+-42+to+-40
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=85052.0;attach=33860;image)
It is not clear yet if it's just an artefact caused by rounding error in the algorithm, or it's an inherent feature of Zeta function. Although, the first case seems to be more likely.
Someone thinks that he might have solved the problemIt seems like he has realized his mistake.
Riemann's hypothesis can be proven by showing that ζ(s) - ζ(1-s*) = 0 if and only if s=1-s*, which implies that Re(s)=1/2We can construct an equation
The statement "The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros" is true, but understanding why requires some familiarity with complex analysis and the properties of the Riemann zeta function. Here's a breakdown:
1. What is the Riemann zeta function?
The Riemann zeta function, denoted by ζ(s), is a function defined for complex numbers 's' (except for 's = 1'). It plays a crucial role in understanding prime numbers and other important areas of mathematics.
2. What are the trivial zeros?
The zeta function has some "obvious" zeros, called trivial zeros, at negative even integers (-2, -4, -6, etc.). These occur because the series definition of the zeta function diverges at these points.
3. What is the functional equation?
The functional equation is a powerful relationship between the zeta function at 's' and its value at 1 - s. It essentially "mirrors" the behavior of the function across the line Re(s) = 1/2 (where Re(s) denotes the real part of 's').
4. How does the functional equation imply no zeros with negative real parts?
Suppose ζ(s) had a non-trivial zero with a negative real part, say at s = -a (where a is positive). By the functional equation, this would imply another zero at s = 1 + a, which has a positive real part.
However, we know that the zeta function only diverges at negative even integers, and it is non-zero for all other positive real values. Therefore, a zero at 1 + a would contradict this property.
This contradiction arises because having a zero with a negative real part essentially "unpairs" the relationship between s and 1 - s through the functional equation. Since such "unpairing" leads to inconsistencies, it follows that such zeros cannot exist.
5. Important note:
This explanation necessarily simplifies a complex mathematical argument. For a rigorous understanding, deeper knowledge of complex analysis and the properties of the zeta function is required.
No, there are no zeros of the Riemann zeta function with real part greater than 1/2. This is a known result in mathematics, but with two levels of "known":
Level 1: Proven results:
For Re(s) > 1: We can prove using the Euler product formula that the zeta function is never zero in this region.
For Re(s) = 1: While not as simple as the first point, it has also been proven that the zeta function does not have any zeros on the line Re(s) = 1.
Level 2: Unsolved but highly likely:
The Riemann Hypothesis: This famous conjecture posits that all non-trivial zeros of the zeta function lie on the "critical line" where Re(s) = 1/2. Although unproven, it is widely believed to be true based on extensive numerical calculations and theoretical evidence.
In conclusion, with high confidence, we can say that there are no zeros of the Riemann zeta function with real part greater than 1/2. If you're interested in learning more about the zeta function and its zeros, I can provide further information or point you towards relevant resources.
Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line s = 1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1. He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.1-s* is the formula reflecting point s with respect to critical line x=1/2.
https://en.wikipedia.org/wiki/Riemann_hypothesis
I'd like to introduce a function, which I call Y function, just because that's how its plot look like in complex plane. It can also called upsilon function, just to sound more geeky. It's defined as follow:I played around with this Y function, and found something interesting when 1 in the formula is replaced with other numbers. The plot looks simpler when s has high imaginary part.
Y(s)=ln{ζ(s) - ζ(1-s*)}
When m =1, the low slope part isn't visible.
I played around with this Y function, and found something interesting when 1 in the formula is replaced with other numbers. The plot looks simpler when s has high imaginary part.
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=85052.0;attach=34306)
The logarithm of a complex number z=log(r*e^(iθ)) is condensed as log(z)=log(r)+iθ, where r is the absolute value of z and θ is the argument of z. It is represented in polar coordinates.
when imaginary part of s is close to the smallest non-trivial zero of Riemann's zeta functionAnother trend is that at larger value of Re(s), imaginary part of Y function has smaller period.
https://www.wolframalpha.com/input?i=plot+%28ln%28%28zeta%28s%2B14.134725+i%29%29+-+%28Zeta%281-s%2B14.134725+i%29%29%29+%29from+-22+to+23
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=85052.0;attach=34292)
Any larger value for the imaginary part of s don't change shape of the real part of Y function, only the imaginary part gets longer period.
https://www.wolframalpha.com/input?i=plot+%28ln%28%28zeta%28s%2B114.134725+i%29%29+-+%28Zeta%281-s%2B114.134725+i%29%29%29+%29from+-22+to+23
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=85052.0;attach=34294)
The Riemann Hypothesis is the most notorious unsolved problem in all of mathematics. Ever since it was first proposed by Bernhard Riemann in 1859, the conjecture has maintained the status of the "Holy Grail" of mathematics. In fact, the person who solves it will win a $1 million prize from the Clay Institute of Mathematics. So, what is the Riemann hypothesis? Why is it so important? What can it tell us about the chaotic universe of prime numbers? And why is its proof so elusive? Alex Kontorovich, professor of mathematics at Rutgers University, breaks it all down in this comprehensive explainer.
00:00 A glimpse into the mystery of the Riemann Hypothesis
01:42 The world of prime numbers
02:30 Carl Friedrich Gauss looks for primes, Prime Counting Function
03:30 Logarithm Function and Gauss's Conjecture
04:39 Leonard Euler and infinite series
06:30 Euler and the Zeta Function
07:30 Bernhard Riemann enters the prime number picture
08:18 Imaginary and complex numbers
09:40 Complex Analysis and the Zeta Function
10:25 Analytic Continuation: two functions at work at once
11:14 Zeta Zeros and the critical strip
12:20 The critical line
12:51 Why the Riemann's Hypothesis has a profound consequence to number theory
13:04 Riemann's Hypothesis shows the distribution of prime numbers can be predicted
14:59 The search for a proof of the Riemann Hypothesis
Why is the Riemann Hypothesis hard? Just one reason (of very many): it's not an analytic question. Here are the values of zeta on the 1/2-line (where at least 40% of the zeros are, all should be) and the 4/5-line, where none should be. The latter gets arbitrarily close to 0, i.o.
https://www.johndcook.com/blog/2019/11/29/near-zeros-of-zeta/
Any idiot can build a bridge that stands, but it takes an engineer to build a bridge that barely stands.
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
Let's focus on the imaginary part of the addition equation.
https://www.wolframalpha.com/input?i=plot+im%28zeta%280.5%2Bis%29+%2B+zeta%280.5-is%29%29+from+0+to+30
And the real part of the subtraction equation.
https://www.wolframalpha.com/input?i=plot+re%28zeta%280.5%2Bis%29+-+zeta%280.5-is%29%29+from+0+to+30
They are always zero.
Compare them to the value of s where the real part is slightly different from 0.5.
https://www.wolframalpha.com/input?i=plot+im%28zeta%280.5001%2Bis%29+%2B+zeta%280.4999-is%29%29+from+0+to+30
https://www.wolframalpha.com/input?i=plot+re%28zeta%280.5001%2Bis%29+-+zeta%280.4999-is%29%29+from+0+to+30
Another result I got from searching.Most previous works tried to find patterns in the plot of Zeta function on imaginary axis.QuoteWhy is the Riemann Hypothesis hard? Just one reason (of very many): it's not an analytic question. Here are the values of zeta on the 1/2-line (where at least 40% of the zeros are, all should be) and the 4/5-line, where none should be. The latter gets arbitrarily close to 0, i.o.
https://www.johndcook.com/blog/2019/11/29/near-zeros-of-zeta/
https://www.wolframalpha.com/input?i=plot+%7Czeta%281%2F2+%2B+s.+i%29%7C+from+0+to+30But they are too chaotic to collect useful patterns or information.
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=85052.0;attach=34296)
https://www.wolframalpha.com/input?i=plot+%7Czeta%281%2F2+%2B+s.+i%29%7C+from+1000+to+1030
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=85052.0;attach=34298)
The plots above show that at higher imaginary part of s, the density of Zeta zero is higher. It reduces the space for anomalous zero. Thus the higher the imaginary part, the less likely we can find anomalous zero of Zeta function. It provides a stronger indication that Riemann's hypothesis is true.
The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros.The equivalent statement for the trivial zero is:
The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:
The real part of every nontrivial zero of the Riemann zeta function is 1/2
Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1/2 + i t, where t is a real number and i is the imaginary unit.
https://en.wikipedia.org/wiki/Riemann_hypothesis
Corrections to an estimate of the prime-counting function using zeros of the zeta function. The magnitude of the correction term is determined by the real part of the zero being added in the correction.Instead of real part, it should say imaginer part.
https://en.m.wikipedia.org/wiki/Riemann_hypothesis
This is the first episode of the RH Saga.
We embark on a journey into the world of L-functions, by introducing the Riemann Hypothesis and the dream of a new geometry over the "field with one element".
The aim of RH Saga Season 1 is to map the landscape of L-functions, as a foundation for future in-depth exploration of some of the most immortal math problems of all time.
This video is part of a PeakMath course. Join the journey at https://www.peakmath.org/
---
Chapters:
00:00 - Introduction to the RH Saga
01:50 - Introduction to Episode 1: The Dream
03:17 - Chapter 1: Intro to F1
07:58 - Summary of Chapter 1
09:26 - Chapter 2: Recap of RH
18:46 - Chapter 3: Proof of RH?
21:40 - Summary of Chapter 3
Links:
1. "Numbers as Functions" - Yuri Manin
https://arxiv.org/pdf/1312.5160.pdf
2. "Riemann's Hypothesis" - Brian Conrey
https://tinyurl.com/conrey90
3. SageMathCell
https://sagecell.sagemath.org/
4. SageMath
https://www.sagemath.org/
5. SageMathCell permalink for Riemann spectrum code
https://tinyurl.com/e1-spectrum
6. SageMathCell permalink for cosine waves code
https://tinyurl.com/e1-cosine
Errata:
Around 6:44: Just to clarify - the "fractional Gamma value" \Gamma(p/q) may not be a period in itself, but \Gamma(p/q)^q is certainly a period.
This is the third episode of the RH Saga.
We continue our journey by considering the zeros of L-functions beyond the Riemann zeta function. The location of these zeros is the subject of the generalized Riemann hypothesis. The story of these zeros also give a first indication of the general relationship between prime numbers and L-functions.
The overall aim of RH Saga Season 1 is to map the landscape of L-functions, as a foundation for future in-depth exploration of some of the most immortal math problems of all time.
Chapters:
00:00 - Intro
02:00 - Review of examples
05:04 - Analytic continuation
12:39 - Zeros in the critical strip
16:44 - Cosine waves
21:35 - Final remarks
---
Links:
1. LMFDB
https://www.lmfdb.org/
2. "The Music of the Primes" on Amazon UK:
https://www.amazon.co.uk/Music-Primes...
---
Errata:
The integral written at approx. 11:00 has a slight mistake. Instead of the number 2, the variable s should be used inside the sine function in the integrand. Correction displayed on screen at 11:20.
---
Riemann HypothesisInterestingly, it says nothing about analytic continuation. Although it provides a link to Official Problem Description by E. Bombieri https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf
The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann?s 1859 paper, it asserts that all the ?non-obvious? zeros of the zeta function are complex numbers with real part 1/2.
(https://www.claymath.org/wp-content/uploads/2022/05/riemann_1859_a_0.jpg)
Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern. However, the German mathematician G.F.B. Riemann (1826 ? 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ? called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation ζ(s) = 0 lie on a certain vertical straight line.
This has been checked for the first 10,000,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.
All this leads to several basic questions.
Is there a theory in the global case, playing the same role as cohomology does
for Zeta functions of varieties over a field of positive characteristic? Is there an
analogue of a Frobenius automorphism in the classical case? Is there a general
index theorem by which one can prove the classical Riemann hypothesis? We are
here in the realm of conjectures and speculation. In the adelic setting propounded
by Tate and Weil, the papers [Conn], [Den], [Hara] offer glimpses of a possible setup
for these basic problems.
On the other hand, there are L-functions, such as those attached to Maass
waveforms, which do not seem to originate from geometry and for which we still
expect a Riemann hypothesis to be valid. For them, we do not have algebraic and
geometric models to guide our thinking, and entirely new ideas may be needed to
study these intriguing objects.
Problems of the Millennium: the Riemann Hypothesis
E. Bombieri
https://www.claymath.org/millennium-problems/rules/
Rules for the Millennium Prize Problems
The revised rules for the Millennium Prize Problems were adopted by the Board of Directors of the Clay Mathematics Institute on 26 September, 2018.
Please read this document carefully before contacting CMI about a proposed solution. In particular, please note that:
CMI does not accept direct submission of proposed solutions.
The document is a complete statement of the rules and procedures: CMI will not offer any futher guidance or advice.
Before CMI will consider a proposed solution, all three of the following conditions must be satisfied: (i) the proposed solution must be published in a Qualifying Outlet (see ?6), and (ii) at least two years must have passed since publication, and (iii) the proposed solution must have received general acceptance in the global mathematics community
Here are some interesting results which might be useful in solving the problem.After consulting with Gemini, it becomes obvious for me that Riemann's functional equation plays an important role to convince that all trivial zeros of Riemann's Zeta function lie on the real line, ie. their imaginary part is zero. I suspect that the same functional equation can somehow convincingly show that all non-trivial zeros of Riemann's Zeta function lie on the critical line, ie. their real part is 1/2.Quotehttps://en.wikipedia.org/wiki/Riemann_zeta_function#Riemann's_functional_equation
This zeta function satisfies the functional equation
(https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c33a4fef2cb6271b95028cf7a012d2c0d0ec00)
where Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product sin(πs/2)Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor.Quotehttps://en.wikipedia.org/wiki/Riemann_zeta_function#Other_results
The fact that
(https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b09034e8f5f6df4406385816e4a6f7109e19cc)
for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = ½
It is also known that no zeros lie on a line with real part 1.Quotehttps://en.wikipedia.org/wiki/Riemann_hypothesis#Zeros_on_the_critical_line
Hardy (1914) and Hardy & Littlewood (1921) showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function. Selberg (1942) proved that at least a (small) positive proportion of zeros lie on the line. Levinson (1974) improved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and Conrey (1989) improved this further to two-fifths. In 2020, this estimate was extended to five-twelfths by Pratt, Robles, Zaharescu and Zeindler[22] by considering extended mollifiers that can accommodate higher order derivatives of the zeta function and their associated Kloosterman sums.Quotehttps://mathworld.wolfram.com/RiemannHypothesis.html
It is known that the zeros are symmetrically placed about the line I(s)=0. This follows from the fact that, for all complex numbers s,
1. s and the complex conjugate s* are symmetrically placed about this line.
2. From the definition (1), the Riemann zeta function satisfies zeta(s*)=zeta(s)*, so that if s is a zero, so is s*, since then zeta(s*)=zeta(s)*=0*=0.
It is also known that the nontrivial zeros are symmetrically placed about the critical line R(s)=1/2, a result which follows from the functional equation and the symmetry about the line I(s)=0. For if s is a nontrivial zero, then 1-s is also a zero (by the functional equation), and then 1-s* is another zero. But s and 1-s* are symmetrically placed about the line R(s)=1/2, since 1-(x+iy)*=(1-x)+iy, and if x=1/2+x', then 1-x=1/2-x'.
From above results, it can be inferred that for any point in the critical strip,
ζ(s*)=ζ(1-s) if and only if s is in critical line,
where s*=complex conjugate of s.
For every nontrivial zero of Zeta function, ζ(s) =0,(https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c33a4fef2cb6271b95028cf7a012d2c0d0ec00)
Re(Y(s)) = - ~ (negative infinity)
Im(Y(s) = undefined (due to switching between two different, discontinuous values)
The plot of Y function suggests that those conditions can only be satisfied where Re(s) = 1/2, as Riemann predicted.
At this point, it should be obvious that direct attack on the problem is impossible.
But it doesn't stop us from trying anyway. We can start from the functional equation.Here are some interesting results which might be useful in solving the problem.When Riemann zeta function produces 0 result, ζ(s) = 0, at least one of these terms is 0Quotehttps://en.wikipedia.org/wiki/Riemann_zeta_function#Riemann's_functional_equation
This zeta function satisfies the functional equation
(https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c33a4fef2cb6271b95028cf7a012d2c0d0ec00)
where Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product sin(πs/2)Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor.
1) 2s = 0 → s = -∞
2) πs-1 = 0 → s = -∞
3) sin(πs/2).Γ(1-s) = 0 → s ∈ {negative even numbers}
4) ζ(1-s) = 0 = ζ(s)
Point #3 gives trivial zeros, while point #4 gives non-trivial zeros.
when 1-s=s → 1=2s → s=1/2
But ζ(1/2) <> 0
The derivation above should be enough to show that Riemann's Hypothesis is true, ie. non-trivial zeros of Riemann's Zeta function has real part of 1/2. While trivial zeros are determined by sine function in the functional equation (minus those cancelled by the Gamma function), non-trivial zeros are determined by analytic continuation of Zeta function to region where the real part is less then 1.From above results, it can be inferred that for any point in the critical strip,
ζ(s*)=ζ(1-s) if and only if s is in critical line,
where s*=complex conjugate of s.4) ζ(1-s) = 0 = ζ(s)The fact thatBut we only care about the case where ζ(s) = 0.
ζ(s) = (ζ(s*))*
0* = 0
→ ζ(s) = ζ(s*) = ζ(1-s) = 0
when s* = 1-s → s+s* = 1
→ Re(s)+Im(s).i + Re(s)-Im(s).i = 1 → 2.Re(s) = 1
→ Re(s) = 1/2
To be clear, I don't dispute the validity that all trivial zeros are on the real line.For zeros of Riemann's Zeta function, log of subtractive symmetry checking functions like Y and M functions necessarily yield negative infinity, but it's still inadequate. They also must yield negative infinity when the functions use multiplication instead of subtraction.