Wednesday, May 22, 2013

A proof of the Riemann Hypothesis using the convergence of an integral

Thursday morning update: After many hours, I decided that there is a critical error in the otherwise cleverly constructed proof. On page 138 (discussing Lemma 3), second part, he says "whence the function converges absolutely" essentially for any $$z$$ with a real positive part. But it seems he hasn't really established that (except for circular reasoning) because if RH is false, and it may be false, the numerator $$|\psi(e^t)-e^t|$$ goes like $$e^{at}$$ for some positive $$a$$ and the region of convergence is shifted by $$a$$. So the "absolute" part of the convergence isn't correctly proven, it seems to me. Maybe it's enough to prove the "ordinary" convergence but I suspect that there could be a similar error in the $$g_1$$ part of Lemma 3, too. Apologies if I am making a mistake.
Some people talk about the proof of "almost twin" prime integers separated by at most 70 million or something like that. I am not terribly excited by this result even if it is true. It's always more interesting to talk about somewhat promising proofs to the Riemann Hypothesis, not only because of the \$1 million that will be given to the first person who solves the old puzzle. Many people have thought that they had a proof but the candidate proofs have always failed so far. So you must understand it is extremely likely that we have another example of a failure here. But I am going to tell you, anyway. It would be great if some readers spend a sufficient time and energy by reading the paper. Please don't be repelled by the idiosyncratic Chinese English. Even I can recognize that it's not how a native speaker would formulate the ideas. ;-)

That's his real name. Today, Hao-cong [first name] Wu [surname] of China sent me his new paper with a somewhat strange title (linguistically)
Showing How to Imply Proving The Riemann Hypothesis (PDF full)
published in the European Journal of Mathematical Sciences. How does the proof work?

It's likely that I won't quite reproduce everything that is needed for the proof in this blog entry even though I may try. Teaching things is the best way to learn them. ;-)

Wu elaborates upon some ideas initiated by Serge Lang, a famous mathematician. But that's the last comment about the sociological context. Now, let us look at the ideas which don't seem to require any esoteric new branches of mathematics.

The proof reduces the Riemann Hypothesis to a claim about the absolute convergence of an integral that is related to the Riemann $$\zeta$$-function in a simple way. Let's roll.

The function that Wu finds more convenient is called $$\psi(x)$$, pronounce "psi of ex". It is related to the Riemann $$\zeta$$-function by the following identities$\eq{ \phi(s) &= -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s} =\sum_p \frac{\log p}{p^s-1}=\\ &= s \int_1^\infty \frac{\psi(x)}{x^{s+1}}\dd x = \frac{s}{s-1}+s\int_1^\infty \frac{\psi(x)-x}{x^{s+1}}\dd x }$ where the sum over $$p$$ goes over the primes $$2,3,5,\dots$$. The first step you should be able to verify if you want to validate Wu's proof is that the identities above are satisfied if $$\psi(x)$$ is defined as the manifestly convergent sum$\psi(x) = \sum_{p^m\leq x} \log p = \sum_{n\leq x} \Lambda(n)$ where $$\Lambda(n)=\log p$$ if $$\exists m\geq 1: \,n=p^m$$ for a prime $$p$$ and otherwise it is set to zero. Note that this $$\psi(x)$$ is defined in such a way that for a large $$x$$, it's expected to be very close to $$x$$ because the "probability to be prime" $$1/\log x$$ is cancelled by the factor $$\log p$$ from the definition of $$\psi(x)$$ – it's close enough already when we allow $$m=1$$ only.

The second step is to realize that the presence of a zero or zeroes of $$\zeta(s)$$ also implies (or would imply) a pole of $$\phi(s)$$, the [minus] "logarithmic derivative of the $$\zeta$$-function", at the same location of the complex plane. To prove the Riemann hypothesis, it is sufficient to prove that $$\phi(s)$$ has no poles for $\frac 12 \lt {\rm Re}(s) \lt 1$ (in the "right half-strip", as I will call it) because the hypothetical "RH-violating" zeroes (and singularities) come in pairs symmetrically distributed relatively to the critical axis $$s=1/2+it$$ for $$t\in\RR$$. Note that $$\phi(s)$$ has a pole (or would have a pole) even for a higher-order zero of $$\zeta(s)$$.

The third step, and it's the only hard one, is to actually prove that one of the integrals involving $$\psi(x)$$ used to calculate $$\psi(s)$$ above$\int_1^\infty \frac{\psi(x)-x}{x^{s+1}}\dd x$ is analytic in the right half-strip so it has no poles over there. Consequently, the $$\zeta$$-function has no zeroes in the right half-strip and, by the left-right symmetry, no zeroes in the left half-strip, either.

Wu reduces the claimed analyticity of the integral above to the absolute convergence (convergence even if the integrand is replaced by its absolute value) and uniform convergence (the speed of convergence may be taken to be $$\varepsilon$$-independent), $$\forall\varepsilon\gt 0$$, of the integral$\int_1^\infty \frac{\psi(x)-x}{x^{3/2+\varepsilon}}\dd x.$ It shouldn't be hard to see that the absolute and uniform convergence of the integral above (here) is enough for the analyticity of the previous integral, and therefore for the absence of the non-trivial zeroes. Note that the exponents $$s+1$$ for $$s$$ in the right half-strip and $$3/2+\varepsilon$$ for a positive $$\varepsilon$$ are the same objects.

So aside from the claims that should be straightforward, the beef of the proof should be the demonstration of the absolute and uniform convergence of the integral in the last displayed equation.

Note that Wu's approach is linked both to the "complex analytic" interpretation of the Riemann Hypothesis as well as the prime-integer-counting, "number-theoretical" interpretation. It's because sufficient experts know that the Riemann Hypothesis is equivalent to the statement$\forall \varepsilon\gt 0: \, \psi(x) = x+ O(x^{1/2+\varepsilon})$ which says that if we accept that the probability for a "rough number $$x$$" to be a prime is $$1/\log(x)$$, then the estimated number of primes up to $$n$$ deviates from the actual one at most by a power law (that is producing the $$O(\dots)$$ term above.

Proving the convergence

OK, so how does Wu want to prove the uniform and absolute convergence? He offers some introduction to the theory of functions of real and complex variables together with some lemmas that are not quite well-known and that may even be new. Finally, the proof boils down to the existence (for any $$s$$ with a real positive part) of the Laplace transforms $$g_{1,2}(s)$$ of a function called $$f_{1,2}(t)$$ related to $$\psi(e^t)-e^t$$ for the subscript $$1$$ or its absolute value for the subscript $$2$$.

If you quickly want to focus on claims related to the $$\zeta$$-function and ignore various theorems and lemmas about completely general functions and their convergence etc. (assuming that these things are harmless and perhaps known to you, explicitly or intuitively), you may find it helpful for me to say that only Theorem 5 (among 7 theorems) and Lemma 3 (among 3 lemmas) is what you want to read. If there is some circularity in Wu's argument (secretly assuming RH), it's probably somewhere in Theorem 5 or Lemma 3.

In particular, I believe that Theorem 5 contains the main trick that allows us to show the convergence in the right half-strip. This theorem claims the absence of poles (except for the $$s=1$$ pole) of the function$\eq{ \Phi(s) &= \sum_p \frac{\log p}{p^s} = \phi(s)-\sum_p h_p(s),\\ |h_p(s)|&\leq B\frac{\log p}{|p^{2s}|} }$ On one hand, this capital $$\Phi(s)$$ is shown to be rather close to the lowercase $$\phi(s)$$, using an argument based on geometric series. On the other hand, the $$2s$$-th power of something appears in the difference between $$\Phi$$ and $$\phi$$ which makes $$\sum\log n/n^{2s}$$ converge for $${\rm Re}(s)\geq 1/2+\delta$$. So the coefficient $$2$$ in $$2s$$ here is the ultimate reason why the meromorphic character of $$\Phi(s)$$ starts at $${\rm Re}(s)\gt 1/2$$, how we get the one-half somewhere, and why the critical axis becomes a decisive boundary for the well-definedness of $$\phi(s)$$, too.

I don't see any mistake so far but I haven't really devoured all the beef of the proof yet, either, so no complete confirmation from your humble correspondent yet. But it is apparently making more sense every minute!

See the previous TRF blog entries mentioning the Riemann Hypothesis.

1. It will be a big feather in China's cap if proven. According to Wikipedia there have been two Chinese Fields medalists but neither was born or trained in China.

2. The IQ figure is invented.

For Fields medals, I assume you mean Terence Tao and Shing Tung Yau, but Tao was born in Australia and never lived in China. S.T. Yau lived in British controlled Hong Kong from the age of 4 months until he moved to the U.S.

3. Or perhaps you're not a trained mathematician in this area and therefore don't know what you're talking about? ;)

4. Hey, Lubos

You and Tommaso Dorigo sound so alike sometimes. Tommaso has the recent cold fusion paper, you have this Riemann hypothesis paper. I wonder if Tommaso will do a tongue in cheek blog of you becoming a math crackpot ;)

5. Brian G ValentineMay 22, 2013, 7:44:00 PM

The proof is not correct and said integral converges if the Riemann hypothesis is true.

Frankly I agree with Littlewood that RH is probably not true, in there is no reason for it to be true. I think counter example will eventually be shown (for some number that will never appear in digit form).

Anyway I think Generalized RH is true, however.

6. Doesn't the Zeta function in this form only make sense for Re(s)>1. Doesn't it require an analytic continuation to the rest of the complex plane to even discuss the strip 1/2<Re(s)<1? Where is the analytic continuation in this proof?

7. A difference is that no one has revealed a reason why this proof is wrong so far.

8. 9. I don't understand what problem with the proof you have found.

10. The RH is just a special case of the GRH, so how can you think the RH is probably not true but the GRH is true??

11. Definitely true.

12. Brian G ValentineMay 22, 2013, 10:09:00 PM

Thank you, I haven't looked at it very closely yet, but it does not appear that to me convergence of that integral is guaranteed unless the Riemann hypothesis is true

More later, thanks!

13. Brian G ValentineMay 22, 2013, 10:41:00 PM

Thanks, I meant to say, I think GRH is true in some other cases (as Piltz wrote it), but not the case for which RH is true.

14. antonio carlos mottaMay 22, 2013, 11:34:00 PM

I THINK THAT DOESN'T EXIST ONE ONLY ONE RESOLUTION TO THE RIEMANN'S HYPOTHESYS.
THERE ARE SEVERAL "HOLES" THAT ARE TO THE SAME THE THEIRS ARE REAL AND COMPLEX VARIABLES.O OPERATOR IMPLIES THE EVOLUTIONS OF THE SYSTEMS.THEN THE "HOLES
IS 1 OR -! SIMULTANEOUSLYmTHAT COULD TO BE MEASURED BY TWO OPPOSED ORIENTATIONS THAT SEND INFORMATIONS TO THE POSITIVE AND THE OTHER DOES IN OPOSED ORIENTATIONS

15. The paper's author has a blog (in Chinese): http://whc8778.blog.163.com/

16. This is a hilariously terrible paper and European Journal of Mathematical Sciences is an execrable journal. A travesty. No, it's a travesty of a mockery of a sham. Lubos, I think you are trolling us just as Tommaso Dorigo is trolling by pretending to take cold fusion seriously. It's not reasonable to expect mathematicians to debunk every crackpot "proof" of RH that comes along. They have better things to do.

17. Dear Jason, I haven't found any clear error in the paper yet - the strategy looks very sensible.

I agree that the journal is not even worth the name but your superficial way of deciding whether something is right or not is a much worse travesty than everything you criticize.

18. I think the 4 color theorem is not a very good example, exactly because of the use of computers, which could not have been done much earlier.

Anyway, it's fun to try to proof something hard even if it's extremely likely to fail.

19. Dear Mark, so let me give you another, computer-free example.

Several days ago, the proof about "almost twin primes" separated by 70,000,000 or less was announced.

In principle, Yitang Zhang's proof is elementary maths, and expert mathematicians say that he nevertheless succeeded where experts have failed.

http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture

http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989

The idea that all important proofs have to be done by well-known experts and the most esoteric and new techniques is just crap. Sometimes it's so, sometimes it's not so.

20. His english is terrible. I don't think a proof so simple exist.

I has just browse the paper and the last sentence in page 137 is wrong. Both substitutions are wrong, an e^t is missing in both integrals on the right.

21. Regarding Perelman's work. There are deep math or what you called advanced tools of course. eg. Ricci flow with surgery, W-entropy, collapsing theory, etc. I don't think these are not advanced tools at least these tools are not accessible to most of the graduate students.

22. elementary maths? are you kidding me? you look just like a crackpot

According to one of the world leading experts on analytic number

theory, Henryk Iwaniec,Yitang Zhang was not well-known to specialists in

number theory before his fantastic paper on prime numbers was recognized by

the Annals of Mathematics three weeks ago. But he possessed the knowledge of

the most sophisticated areas of analytic number theory, and he could use it

all with ease. Also, he was able to make a breakthrough where many

investigators were stuck, not because something little was overlooked, but

because of new, clever arrangements which he introduced and brilliantly

executed. You could sense immediately that the work had a great chance to be

fine by looking at the clear and logical architecture of the arguments. It

does not mean the paper is simple or elementary. To the contrary, the work

of Zhang constitutes the state-of-the art of analytic number theory. It also

borrows gracefully from other areas, for example, it makes use indirectly

of the Riemann hypothesis for varieties over finite field. Zhang's work will

trigger a lasting avalanche of refinements and improvements with many

innovations. Overnight Zhang redirected the focus of analytic number theory.

How long do we need to wait to see what comes next?"

23. Dear Petr, be sure that I have read everything here and please don't lie. But just because I have read something doesn't imply that I am going to uncritically parrot it or misinterpret it.

I have explained what I meant by the adjective "elementary", I insist it is the right description of the proof, and I also insist that the bulk of the Wu's proof uses equally elementary or non-elementary tools as the bulk of Zhang's proof.

24. I think whether or not OVERSPECIALIZED, the world-class mathematicians have a better taste in math than you do. Also since you said that "One only has one life to live", I think you wasted too much time in Wu Haosong's paper. Wu is a Chinese salesman and do not have any academic training. He is a crackpot and EJMS is a junk journal. Instead Annals of Mathematics is the most prestigious mathematics journals and Zhang earned PhD from Purdue Univ.

I don't think these two things have any similarities. You put them together only reflect your shallowness.

25. Zhang's research is groundbreaking. Please publish some papers on String Theory before you make such naive judgements on other people's work. Tell me bro, what have you contributed to the science world lately?

26. are you working on RH like Perelman?

27. Brian G ValentineMay 30, 2013, 5:00:00 AM

I think you're right Lubos, and from what I see, he has already assumed absolute and uniform convergence of his integral to claim the existence of the Laplace transform of it at all.

So it looks like we have a demonstration that the Riemann hypothesis is true if the Riemann hypothesis is true (I believe John Horton Conway used to collect these as a hobby).

Finding the cases where RH is false numerically might be impossible if the computation involves functions like loglog(n); what are we up to now? Something to the order E+12 if I am not mistaken, that is a nothing to the function loglog(n).

Finding better bounds between the zeros, and bounds for the height of the "spikes" between them might be a way to demonstrate the existence of a contradiction

People have claimed "havoc" if RH is not true - nonsense - people know what the distribution of primes looks like in the event RH is not true, it just means that life isn't as simple as one would like it. Hardy and others proved that some 40% of the nontrivial zeros are real; so maybe about half of them are real, then the other half are imaginary, that's life with a transcendental function.

28. Brian G ValentineMay 30, 2013, 5:18:00 AM

By the way the best part of his paper is his references. Serge Lang presents analysis from the knowledge of an algebraist, everybody ought to run out and buy and read Serge Lang's books right away.

29. http://ejmathsci.org/index.php/ejmathsci/article/view/112/22

30. Sorry, this is exactly the same link that is used in the blog entry.

31. Hello, there's a claimed proof of this Riemann Hypothesis at http://arxiv.org/abs/0809.5120 by Arne Bergstrom. I haven't found anywhere online which correctly claims an error in the proof. Do you mind taking a look?

32. Several months later we have a much clearer picture what Zhang really needed to obtain his result. For one thing, one can get without the Deligne's results on Weil conjectures. Nevertheless, this work is built upon the results of Goldston, Pintz & Yildirim. While these results may fit Luboš's definition of elementary (After all it's just some inequalities and sieves, right?) for most mathematicians are these results rather far from (their definition of) elementary. For interested readers I suggest checking out Polymath8 paper which should be published shortly.

33. I Prove the Riemann Hyporhesis.