I made this visualization of the zeta function in javascript, it is infinitely zoomable, and you can play around with parameters: https://amirhirsch.com/zeta/index.html
It might help you understand why the hypothesis is probably true. It renders the partial sums and traces the path of zeta.
In my rendering, I include all partial sums up to an automatically computed "N-critical" which is when the phase difference between two summands is less than pi (nyquist limit!), after which the behavior of the partial sums is monotonic. The clusters are like alias modes that go back and forth when the instantaneous frequency of the summands is between kpi and (k+1)pi, and the random walk section is where you only have one point per alias-mode. The green lines highlight a symmetry of the partial sums, where the clusters maintain symmetry with the random walk section, this symmetry is summarized pretty well in this paper: https://arxiv.org/pdf/1507.07631
I formed an intuitive signal processing interpretation of the Riemann Hypothesis many years ago, which I'll try to summarize briefly here. You can think of the Zeta function as a log-time sampler -- zeta(s) is the Laplace transform of sum(delta(t-ln n)) which samples at time t=(ln n) for integers n>0, a rapidly increasing sample rate. You can imagine this as an impulse response coming from a black box, and the impulse response can either be finite in energy or a power signal depending on the real parameter.
If you suppose that the energy sum(|1/s|^2) is finite (ie real(s) > 1/2), then the Riemann Hypothesis implies that the sum is non-zero. It is akin to saying that the logarithmic sampler cannot destroy information without being plugged-in.
James Maynard appears regularly on Numberphile so if you'd like to hear some accessible mathematics from one of the authors of this paper I suggest you check it out:
For anyone looking for an introduction to the Riemann Hypothesis that goes deeper than most videos but is still accessible to someone with a STEM degree I really enjoyed this video series [1] by zetamath.
I understood everything in Profesor Tao's OP up to the part about "controlling a key matrix of phases" so the videos must have taught me something!
Trying to imagine what it must feel like to have Terence Tao summarize your argument while mentioning that he'd tried something similar but failed.
"The arguments are largely Fourier analytic in nature. The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them; but they do a number of clever and unexpected maneuvers."
I haven't met him personally, but Tao's writing is very humble and very kind. He talks openly about trying things and not having them work out. And he writes in general a lot about tools and their limitations. I definitely recommend reading his blog.
It must feel like meritocracy. Like when ranking, particularly in strict order, is not the norm - so Terrence Tao doesn't see himself "on top" of anything. Moreover it must imply some solid grounding and a good understanding of how someone's actions are not expected to be correlated with their reputation. This is especially the case where getting the results is a personal or strictly team effort, not a popularity contest.
It can be unexpeted for anyone that's operating in the regular business, corporate, VC and general academic landscape where politics rule while meritocracy is a feel good motivator while popularity is the real coin.
Also curious about the potential significance of a proof. The article is vague:
> (primes) are important for securing encrypted transmissions sent over the internet. And importantly, a multitude of mathematical papers take the Riemann hypothesis as a given. If this foundational assumption were proved correct, “many results that are believed to be true will be known to be true,” says mathematician Ken Ono of Emory University in Atlanta. “It’s a kind of mathematical oracle.”
Are there some obvious, known applications where a RH proof would have immediate practical effects? (beyond satisfaction and 'slightly better encryption').
What are your opinions of all the theorems that rely on RH as an excluded middle?
Constructivists reject exmid, saying instead that a proof of "A or B" requires you to have in hand a proof of A or a proof of B. And nobody yet has a proof of RH nor a proof of ~RH. This is important in so-called incomplete logical systems, where some theorems are neither provable nor disprovable, and, therefore, exmid is an inadmissible axiom.
This comments section is very oddly filled with people who don't actually understand the subject matter but want to sound smart, and then accomplish the opposite. Let go of your insecurities people, it's ok to not understand some things and be open about it. Everyone doesn't understand more things than they do.
Apart from one flagged comment I find the comments to be quite profound and interesting, we even have a cool visualization demo of the Riemann zeta function:
This comments section is very oddly filled with people who don't actually understand the subject matter but want to sound smart, and then accomplish the opposite.
One of the most important open problems in mathematics is called the Riemann hypothesis. It states that the solutions of a certain equation `zeta(z)=0` are all of a particular type. Almost every living mathematician has tried to solve it at some point in their lives. The implications of the hypothesis are deep for the theory of numbers, for instance for the distribution of prime numbers.
In a recent paper some mathematicians claim they have put some stronger bounds on where those solutions can be. In this link Terrence Tao, one of the most acclaimed mathematicians alive speaks very highly about the paper.
IMHO, this is probably not of huge interest to not mathematicians just yet. It is an extremely technical result. And pending further review it might very well be wrong or incomplete.
There are lots of places you can read about the Riemann Hypothesis, its implications and its attempts to solve it.
We have an approximate expression of how many prime numbers there are less than N, as N gets larger. If the Riemann hypothesis is true, then we know that the errors in this approximation are nice and small, which would allow us to prove many other approximate results. (There are many results of the form "If the Riemann hypothesis is true, then...")
I think I read somewhere ("The Music of the Primes" maybe?) that the Riemann Hypothesis says that all zeroes of the Zeta function are on a line in the complex plane, and that although it is unproved, it has been proved all the zeroes are within a narrow strip centred on the line and you can make the strip as arbitrarily narrow as you like. I often wonder if I misunderstood somehow because it seems to me if it is really as I just restated it, well the Riemann Hypothesis is obviously true, that proof is "good enough" for engineering purposes anyway.
"It has been proved all the zeroes are within a narrow strip centred on the line and you can make the strip as arbitrarily narrow as you like."
Nothing close to this is known.
The nontrivial zeros of zeta lie within the critical strip, i.e., 0 <= Re(s) <= 1 (in analytic number theory, the convention, going back to Riemann's paper is to write a complex variable as s = sigma + it)*. The Riemann Hypothesis states that all zeros of zeta are on the line Re(s) = 1/2. The functional equation implies that the zeros of zeta are symmetric about the line Re(s) = 1/2. Consequently, RH is equivalent to the assertion that zeta has no zeros for Re(s) > 1/2. A "zero-free region" is a region in the critical strip that is known to have no zeros of the Riemann zeta function. RH is equivalent to the assertion that Re(s) > 1/2 is a "zero-free region." The main reason that we care about RH is that RH would give essentially the best possible error term in the prime number theorem (PNT) https://en.wikipedia.org/wiki/Prime_number_theorem. A weaker zero-free region gives a weaker error term in the PNT. The PNT in its weakest, ineffective form is equivalent to the assertion that Re(s) >= 1 is a zero free region (i.e., that there are no zeros on the line Re(s) = 1).
I think your confusion stems from the fact that approximately the reverse of what you said above is true. That is, the best zero-free regions that we know get arbitrarily close to the Re(s) = 1 line (i.e., get increasingly "useless") as the imaginary part tends to infinity. Your statement seems to suggest that the the area we know contains the zeros gets arbitrarily close to the 1/2 line (which would be amazing). In other words, rather than our knowledge being about as close to RH as possible (as you suggested), our knowledge is about as weak as it could be. (See this image: https://commons.wikimedia.org/wiki/File:Zero-free_region_for.... The blue area is the zero-free region.)
* I don't like this convention; why is it s = sigma + it instead of sigma = s + it? Blame Riemann.
Does he have any beginner friendly books on math I saw some clips of his masterclass but I am skeptical of masterclass in general so I am not sure about that
This is pure math, so not much, at least directly or immediately.
But I find it amusing how this argument always comes up and how it goes back millenia.
A student of Plato (428 B.C. -- 348 B.C.) once asked the great master, "What practical uses do these theorems serve? What is to be gained from them?" Plato's answer was immediate and peremptory. He turned to one of his slaves and said, "Give this young man an obol [a small Greek coin] so that he may feel that he has gained something from my teachings. Then expel him."
They had a good bound of the long-tail distribution when x>=3/5=0.6. Now someone extended that result to x>=13/25=0.52. (The long term objective is to prove a stronger version for x>1/2=.5.)
Without intending to take anything away at all from the work of the researchers, I think “breakthrough” might just be overselling it.
They have improved a bound but I can’t tell that their method will open up a plausible path to a full proof. It feels like we’ve managed to increase the top speed of a car from 200mph to 227mph, but are no closer to understanding how the engine works.
[+] [-] amirhirsch|1 year ago|reply
It might help you understand why the hypothesis is probably true. It renders the partial sums and traces the path of zeta.
In my rendering, I include all partial sums up to an automatically computed "N-critical" which is when the phase difference between two summands is less than pi (nyquist limit!), after which the behavior of the partial sums is monotonic. The clusters are like alias modes that go back and forth when the instantaneous frequency of the summands is between kpi and (k+1)pi, and the random walk section is where you only have one point per alias-mode. The green lines highlight a symmetry of the partial sums, where the clusters maintain symmetry with the random walk section, this symmetry is summarized pretty well in this paper: https://arxiv.org/pdf/1507.07631
[+] [-] amirhirsch|1 year ago|reply
If you suppose that the energy sum(|1/s|^2) is finite (ie real(s) > 1/2), then the Riemann Hypothesis implies that the sum is non-zero. It is akin to saying that the logarithmic sampler cannot destroy information without being plugged-in.
[+] [-] atonalfreerider|1 year ago|reply
Mine is in Unity and shows the spiral in 3D, up the Y axis. I think it's helpful to see in three dimensions: https://github.com/atonalfreerider/riemann-zeta-visualizatio...
[+] [-] md224|1 year ago|reply
Still, it's funny to see how many people have attempted this! It's a fun programming exercise with a nice result.
[+] [-] nyc111|1 year ago|reply
[+] [-] 3abiton|1 year ago|reply
[+] [-] nickcw|1 year ago|reply
https://www.youtube.com/playlist?list=PLt5AfwLFPxWJdwkdjaK1o...
[+] [-] nomilk|1 year ago|reply
Source: https://www.youtube.com/watch?v=eupAXdWPvX8&list=PLt5AfwLFPx...
[+] [-] ks2048|1 year ago|reply
[+] [-] samscully|1 year ago|reply
I understood everything in Profesor Tao's OP up to the part about "controlling a key matrix of phases" so the videos must have taught me something!
[1] https://www.youtube.com/watch?v=oVaSA_b938U&list=PLbaA3qJlbE...
[+] [-] idlewords|1 year ago|reply
"The arguments are largely Fourier analytic in nature. The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them; but they do a number of clever and unexpected maneuvers."
[+] [-] nybsjytm|1 year ago|reply
[+] [-] ants_everywhere|1 year ago|reply
[+] [-] mi_lk|1 year ago|reply
[0]: https://en.wikipedia.org/wiki/Larry_Guth
[1]: https://en.wikipedia.org/wiki/James_Maynard_(mathematician)
[+] [-] random3|1 year ago|reply
It can be unexpeted for anyone that's operating in the regular business, corporate, VC and general academic landscape where politics rule while meritocracy is a feel good motivator while popularity is the real coin.
[+] [-] unknown|1 year ago|reply
[deleted]
[+] [-] pkilgore|1 year ago|reply
[1] https://www.sciencenews.org/article/why-we-care-riemann-hypo...
[+] [-] nomilk|1 year ago|reply
> (primes) are important for securing encrypted transmissions sent over the internet. And importantly, a multitude of mathematical papers take the Riemann hypothesis as a given. If this foundational assumption were proved correct, “many results that are believed to be true will be known to be true,” says mathematician Ken Ono of Emory University in Atlanta. “It’s a kind of mathematical oracle.”
Are there some obvious, known applications where a RH proof would have immediate practical effects? (beyond satisfaction and 'slightly better encryption').
[+] [-] nsoonhui|1 year ago|reply
[+] [-] cvoss|1 year ago|reply
Constructivists reject exmid, saying instead that a proof of "A or B" requires you to have in hand a proof of A or a proof of B. And nobody yet has a proof of RH nor a proof of ~RH. This is important in so-called incomplete logical systems, where some theorems are neither provable nor disprovable, and, therefore, exmid is an inadmissible axiom.
[+] [-] nerdponx|1 year ago|reply
[+] [-] riemann12|1 year ago|reply
If RH is unprovable one way or another, then certainly no counterexample can exist to the RH otherwise you could find it and prove the RH to be false.
Hence if RH is unprovable, it must be true. I suppose this uses logic outside the logical system that RH operates in.
[+] [-] greenthrow|1 year ago|reply
[+] [-] Maxatar|1 year ago|reply
https://news.ycombinator.com/item?id=40571995#40576767
Your comment is rather condescending and kind of feels like a form of projection rather than a meaningful contribution.
[+] [-] CyberDildonics|1 year ago|reply
Just this one?
[+] [-] tucnak|1 year ago|reply
[+] [-] wing-_-nuts|1 year ago|reply
[+] [-] nhatcher|1 year ago|reply
In a recent paper some mathematicians claim they have put some stronger bounds on where those solutions can be. In this link Terrence Tao, one of the most acclaimed mathematicians alive speaks very highly about the paper.
IMHO, this is probably not of huge interest to not mathematicians just yet. It is an extremely technical result. And pending further review it might very well be wrong or incomplete.
There are lots of places you can read about the Riemann Hypothesis, its implications and its attempts to solve it.
[+] [-] chx|1 year ago|reply
[+] [-] dougbrochill|1 year ago|reply
[+] [-] QuesnayJr|1 year ago|reply
[+] [-] senderista|1 year ago|reply
[+] [-] mr-roboto|1 year ago|reply
[+] [-] billforsternz|1 year ago|reply
[+] [-] gavagai691|1 year ago|reply
Nothing close to this is known.
The nontrivial zeros of zeta lie within the critical strip, i.e., 0 <= Re(s) <= 1 (in analytic number theory, the convention, going back to Riemann's paper is to write a complex variable as s = sigma + it)*. The Riemann Hypothesis states that all zeros of zeta are on the line Re(s) = 1/2. The functional equation implies that the zeros of zeta are symmetric about the line Re(s) = 1/2. Consequently, RH is equivalent to the assertion that zeta has no zeros for Re(s) > 1/2. A "zero-free region" is a region in the critical strip that is known to have no zeros of the Riemann zeta function. RH is equivalent to the assertion that Re(s) > 1/2 is a "zero-free region." The main reason that we care about RH is that RH would give essentially the best possible error term in the prime number theorem (PNT) https://en.wikipedia.org/wiki/Prime_number_theorem. A weaker zero-free region gives a weaker error term in the PNT. The PNT in its weakest, ineffective form is equivalent to the assertion that Re(s) >= 1 is a zero free region (i.e., that there are no zeros on the line Re(s) = 1).
The best-known zero-free region for zeta is the Vinogradov--Korobov zero-free region. This is the best explicit form of Vinogradov--Korobov known today https://arxiv.org/abs/2212.06867 (a slight improvement of https://arxiv.org/abs/1910.08205).
I think your confusion stems from the fact that approximately the reverse of what you said above is true. That is, the best zero-free regions that we know get arbitrarily close to the Re(s) = 1 line (i.e., get increasingly "useless") as the imaginary part tends to infinity. Your statement seems to suggest that the the area we know contains the zeros gets arbitrarily close to the 1/2 line (which would be amazing). In other words, rather than our knowledge being about as close to RH as possible (as you suggested), our knowledge is about as weak as it could be. (See this image: https://commons.wikimedia.org/wiki/File:Zero-free_region_for.... The blue area is the zero-free region.)
* I don't like this convention; why is it s = sigma + it instead of sigma = s + it? Blame Riemann.
[+] [-] greekanalyst|1 year ago|reply
Not only is he one of the greatest mathematical minds alive (if not the greatest), he is also one of the most eloquent writers on mathematics.
Nothing more beautiful than seeing great science being married with great writing.
Even if you don't understand the specifics, you can always get the big picture.
Thank you, Terence!
[+] [-] alexander2002|1 year ago|reply
[+] [-] vsnf|1 year ago|reply
[+] [-] throw_pm23|1 year ago|reply
But I find it amusing how this argument always comes up and how it goes back millenia.
[+] [-] breck|1 year ago|reply
[+] [-] blackle|1 year ago|reply
[+] [-] GalaxyNova|1 year ago|reply
[+] [-] ganzuul|1 year ago|reply
[+] [-] gus_massa|1 year ago|reply
They had a good bound of the long-tail distribution when x>=3/5=0.6. Now someone extended that result to x>=13/25=0.52. (The long term objective is to prove a stronger version for x>1/2=.5.)
[+] [-] lupire|1 year ago|reply
[+] [-] gtsnexp|1 year ago|reply
[+] [-] brcmthrowaway|1 year ago|reply
It makes no sense if the rest of the universe relies upon simple rules
[+] [-] Havoc|1 year ago|reply
[+] [-] jl6|1 year ago|reply
They have improved a bound but I can’t tell that their method will open up a plausible path to a full proof. It feels like we’ve managed to increase the top speed of a car from 200mph to 227mph, but are no closer to understanding how the engine works.