Thursday, March 26, 2009 ... / / Jon Breslaw: a Riemann Hypothesis proof speed contest

The proof is wrong, see the last paragraph if it is the only reason why you are here. The paper has been revised (v3), trying to say more modest things that may be (partly?) true. Jonathan wants me to organize a contest. Jon Breslaw, a Montreal economist, claims to have an elementary proof of the Riemann Hypothesis:
An analytical proof of the Riemann Hypothesis (PDF)
Your task is to be the fastest reader who will explain why the proof is wrong. If it is correct, you should write that you have verified it. ;-)

While I spent an hour in the morning with the old Hilbert-Pólya strategy - finding an explicitly Hermitean operator - a "seemingly random matrix" - whose eigenvalues are "i(s-1/2)" where "s" are the nontrivial roots of the zeta function - Breslaw claims to have a rudimentary proof of a similar kind that I have previously tried but failed.

He claims to have proved a stronger result, namely that the absolute value of the zeta function of "s" is decreasing as a function of the real part of "s" whenever the real part of "s" is between "0" and "1/2". It would follow that "zeta(s)" can't have any roots in the bulk of this half-strip, and RH has to hold.

The statement from the previous paragraph is the meaning of the last Lemma 4.10 that only depends on Lemmas 3.5, 3.6, and 4.7.

At 5:33 PM: Sorry, I won

Sorry, dear readers, but I won. ;-) About five minutes were needed to kill this nonsense.

The crucial Lemma 4.10 in the version 1 of the paper is, as expected, completely ludicrous. For example, even on the real axis, "abs(zeta(s))" is not a decreasing function in the half-strip. For example, "zeta(0)" equals "-1/2" (exactly) while "zeta(1/2)" equals "-1.46" or so which has a larger absolute value than "-1/2". So the absolute value can't be decreasing over there. It is actually strictly increasing on this interval.

I am probably not going to look for the reason why Breslaw thought that he had proved this wrong result - because if he writes statements that can be so easily seen to be wrong, without spending 20 seconds with the most elementary tests, it would be a waste of time.

More generally, I am also pretty certain that no elementary proof that is based on inequalities involving the absolute value of the zeta function can exist. Whether or not the absolute value vanishes and/or its derivative changes its sign depends on the zeta function's zeros themselves. You need to know where the poles are located to complete such a "proof". One can see that assuming that "zeta(s)" has a pair of poles at 0.4 or 0.6 plus "i" times a googolplex, nothing in the "proofs" could ever contradict this assumption.

The zeta function depends on the distribution of the primes that happens to have "no statistical conspiracies" if RH holds. But a proof that knows nothing about the specific features of the zeta function - or the distribution of primes - is simply unlikely (or impossible) to exist because the very result does depend on these things.

In the slow comments, I explain another, "major" mistake in Breslaw's "proof" that cannot be corrected by any small modifications of the paper: in the last "big" sentence of the paper, a contribution of a hypothetical pair of zeros to the derivative is claimed to be negative. But it is very clear that it would actually be positive just on the right side of the left root, being dominated by a pole. Jon Breslaw has agreed that this was a bug that killed the proof claim in v1,v2 and submitted v3 of his paper that doesn't claim to have a proof.

Another "proof"

To make things funnier, there is one more "proof" of the Riemann Hypothesis on the arXiv today. The first page of the preprint by Raghunath Acharya is dedicated to a cartoon by a popular Czech cartoonist, Mr Jiránek. The Czech caption is included and I wonder how the Indian-sounding author could translate it from Czech to English. ;-)

This paper is much less comprehensible than the previous one.

The only place in the paper that seems to be relevant for the Riemann Hypothesis says, in capital letters, that its validity "automatically" follows from a promising yet incomplete approach by Khuri (2001) based on scattering amplitudes in quantum mechanics. Well, I can't say what's exactly wrong with the new Acharya paper but I surely don't believe that RH "automatically" follows from a previous work that was must more complicated and complete than Acharya's rudimentary paper (but realized it wasn't a complete proof).

I am not going to spend more time with this stuff. It seems to be a waste of time. This Acharya - not to be confused with Bobby Acharya of M-theory - tried to attack the other 1-million Clay problems like the QCD mass gap, too. I suspect that both authors are actually motivated by the money (not surprising for the economist, Jon Breslaw) but they have really no clue about the maths and the difficulty of this old problem.    snail feedback (6) :

Re:

zeta(s))" is not a decreasing function in the half-strip. For example, "zeta(0)" equals "-1/2" (exactly) while "zeta(1/2)" equals "-1.46" or so which is a larger absolute value than "-1/2". So the absolute value can't be decreasing over there.

Under lemma 3.2, the condition is that t, (the imaginary part of s) exceeds 2*Pi + del. So that's why it doesn't work at t = 0

Jon Breslaw

Fine, jon, so you've replaced your paper v1 by v2, in order to deal with my criticism. Incidentally, it is bizarre, if I don't use the word "unethical", that you don't acknowledge me as a source.

So the contest is renewed, too, and the readers should look at the validity of v2 with the additional inequality.

By the way, you haven't changed your abstract so the abstract is still wrong.

By the way, your new Lemma 4.10 still fails the one-minute test. You don't quite define Delta but I suppose you mean that it is an arbitrary positive number.

But for the imaginary part equal to 6.284, which strictly exceeds 2 pi, the absolute value of the zeta function is:

Abs[Zeta[0.47 + I 6.284]] = 0.95617

while

Abs[Zeta[0.50 + I 6.284]] = 0.95618.

It's a close call but you can see that it is increasing in that interval between 0.47 and 0.50 for the real part. ;-)

Dear Jon, even if you added the large amount of missing inequalities you have to assume, the "beef" of your proof is still wrong.

What's the key mistake that you won't be able to fix? Well, it's in the last sentence of the last big paragraph. The sentence says that "the total contribution of the pair will still be negative".

That's, of course, bullshit. If "s" is just slightly to the right of "rho_l", then the contribution "Re(1/(s-rho_l))" from "rho_l" is the absolutely dominant one (there is a pole there!) and it is positive, not negative, while other contributions such as one from "rho_r" are negligible. Did you forget to invert "s-rho"?

More generally, it must be very clear to every sane person that no proof that uses the Hadamard product over the zeroes - but doesn't prove anything nontrivial about the configuration of these zeros - can ever exist.

It's very clear that I can construct zeta-like functions with the same s vs. 1-s symmetries and the Hadamard product for them, but the RH will be false for them, so every proof that every such function satisfies RH must be wrong.

Don't be silly.

reader Samuel W Gilbert said...

Here is an excerpt from my new book:

The Riemann Hypothesis & the Roots of the Riemann Zeta Function

by Samuel W. Gilbert

available from amazon.com

http://www.riemannzetafunction.com

© U. S. Copyrights - 2009, 2008, 2005

This book is concerned with the geometric convergence of the Dirichlet series representation of the Riemann zeta function at its roots in the critical strip. The objectives are to understand why non-trivial roots occur in the Riemann zeta function, to define the roots mathematically, and to resolve the Riemann hypothesis.

The Dirichlet infinite series parts of the Riemann zeta function diverge everywhere in the critical strip. Therefore, it has always been assumed that the Dirichlet series representation of the zeta function is useless for characterization of the roots in the critical strip. In this work, it is shown that this assumption is completely wrong.

The Dirichlet series representation of the Riemann zeta function diverges algebraically everywhere in the critical strip. However, the Dirichlet series representation does, in fact, converge at the roots in the critical strip ̵and only at the roots in the critical strip in a special geometric sense. Although the Dirichlet series parts of the zeta function diverge both algebraically and geometrically everywhere in the critical strip, at the roots of the zeta function, the parts are geometrically equivalent and their geometric difference is identically zero.

At the roots of the Riemann zeta function, the two Dirichlet infinite series parts are coincidently divergent and are geometrically equivalent. The roots of the zeta function are the only points in the critical strip where infinite summation and infinite integration of the terms of the Dirichlet series parts are geometrically equivalent. Similarly, the roots of the zeta function with the real part of the argument reflected in the critical strip are the only points where infinite summation and infinite integration of the terms of the Dirichlet series parts with reflected argument are geometrically equivalent.

Reduced, or simplified, asymptotic expansions for the terms of the Riemann zeta function series parts at the roots, equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument at the roots, constrain the values of the real parts of both arguments to the critical line. Hence, the Riemann hypothesis is correct.

At the roots of the zeta function in the critical strip, the real part of the argument is the exponent, and the real and imaginary parts combine to constitute the coefficients of proportionality in geometrical constraints of the discrete partial sums of the series terms by a common, divergent envelope.

Values of the imaginary parts of the first 50 roots of the Riemann zeta function are calculated using derived formulae with 80 correct significant figures using a laptop computer. The first five imaginary parts of the roots are:

14.134725141734693790457251983562470270784257115699243175685567460149963429809256…
21.022039638771554992628479593896902777334340524902781754629520403587598586068890…
25.010857580145688763213790992562821818659549672557996672496542006745092098441644…
30.424876125859513210311897530584091320181560023715440180962146036993329389333277…
32.935061587739189690662368964074903488812715603517039009280003440784815608630551…

It is further demonstrated that the derived formulae yield calculated values of the imaginary parts of the roots of the Riemann zeta function with more than 330 correct significant figures.

continued…

reader Jose Javier Garcia Moreta said...

http://es.scribd.com/doc/98172483/A-FUNCTIONAL-DETERMINANT-FOR-THE-RIEMANN-XI-FUNCTION

here it is a Hamiltonian whose Energies are the square of the imaginary part of the zeros and the Xi function is just a determinant of this Hamiltonian

for the potential see equation (27) (28) and (40) (39) to see how the inverse of the potential is related to the Riemann Weil explicit formula , and the potential is REAL