[hamdani yusuf (OP)]

I just found a great video about prime numbers I'd like to share here.

A Formula for the Primes: Willans' Formula

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?

[hamdani yusuf (OP)]

And here is the latest news regarding Riemann Hypothesis.

https://pandaily.com/mathematician-yitang-zhang-claims-to-have-proven-riemann-hypothesis-problem/

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.

[hamdani yusuf (OP)]

Here's another interesting video on Riemann's Hypothesis.

The Riemann Hypothesis feat. prime numbers

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.

[hamdani yusuf (OP)]

This video proves the Riemann hypothesis. Otherwise, you get a $10,000 reward. (M1)

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.

Just in case you have time to spare.

[hamdani yusuf (OP)]

And here is the sequel.

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.

[hamdani yusuf (OP)]

Here is the start of the video series.

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.

[Zer0]

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?

[hamdani yusuf (OP)]

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.

Some people argued that it won't be practically useful for calculating large prime numbers.

[Zer0]

So in simple words, there is No simple explanation for RH.
Okay.

If i Request a Mathematician to provide me with a list of Prime Numbers uptil 3 Million...

Would they use a pen & paper and take a year to do it?

Or would they use a calculator/computer and give answer in a day?

(Pls don't say they will search online & copy/paste & give a printout within minutes)

[hamdani yusuf (OP)]

(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.

[hamdani yusuf (OP)]

Has anyone noticed a peculiar thing around zeta(-41)?
The real part looks normal.
https://www.wolframalpha.com/input?i=plot%28%28zeta+%28x%29%29%29+from+-42+to+-40


But there's a peculiar jump in imaginary part, as seen in the plot below.
https://www.wolframalpha.com/input?i=plot%28im%28zeta+%28x%29%29%29+from+-42+to+-40

[Zer0]

(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?

[Eternal Student]

Hi.

Post #43  About recommending video(s).

    They have no significant risk of needing to make the payment  - but there is a small question, let's call it a theory, as to whether they even had the intention of making the payment.   That's a question we may never know the answer to.  It is hard to prove and no more likely than the Riemann Hypothesis.

Best Wishes.

[hamdani yusuf (OP)]

(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?

Have you watched this video?

I just found a great video about prime numbers I'd like to share here.

A Formula for the Primes: Willans' Formula

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?

[hamdani yusuf (OP)]

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.
https://www.wolframalpha.com/input?i=plot%28arg%28zeta+%28x%29%29%29+from+-100+to+-0


You can also find jump in imaginary part of zeta(0)

[hamdani yusuf (OP)]

When a small imaginary number is added to the plot, we get different peculiarity.
https://www.wolframalpha.com/input?i=plot%28im%28zeta+%28x%2Bi%2F10e13%29%29%29+from+-42+to+-40
 
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.

[Zer0]

I will check Internet & watch Videos & try my Best to Understand RH.
Until then, Peace Out!

[hamdani yusuf (OP)]

https://youtube.com/playlist?list=PLbaA3qJlbE93DiTYMzl0XKnLn5df_QWqY
This YouTube playlist is a good starting point. A critical information there is about analytic continuation.

[hamdani yusuf (OP)]

Riemann Hypothesis and zeta function interesting visualization on x-axis

[hamdani yusuf (OP)]

Just found another YouTube channel who made exciting math videos. Some materials here may be helpful to our understanding of Riemann zeta function.

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