An unpublished paper by Tribikram Pati, in which he claims to
have disproved the Riemann Hypothesis. I don't have the number
theory background to review it, but I gather the author is a
respected mathematician, possibly retired, from Allahabad, India.
I wondered if any of you could give opinions on it.
interesting, i thought everyone was expecting the opposite.
Did he also include a list of the complex zero's not on the critical
Wait until his paper is vetted by the professionals.
One would assume that would settle the issue, but this paper does not do
that as far as I could tell by glancing at it. The author constructs a
chain of conclusions from the assumption (used essentially throughout)
that the hypothesis is true, ending in something that is supposed to be
obviously false. No construction of other zeros; it fits the word
"disproof" in the title.
It will be interesting to see what experts have to say about this.
David L. Johnson
"Business!" cried the Ghost. "Mankind was my business. The common
welfare was my business; charity, mercy, forbearance, and
benevolence, were, all, my business. The dealings of my trade were but
a drop of water in the comprehensive ocean of my business!" --Dickens,
on p.12, m stands for the multiplicity of a zero of the zeta function,
typically m=1. But then the step from the third to the fourth
something + A < something,
for a positive constant A. Similarly, I do not understand how the
constant B disappears a few lines later. I did not attempt to
though, whether this is a major problem or not.
He explains a couple of pages back that "A" stands for some unspecified
positive constant, which might not be the same constant each time it
appears, even in consecutive lines of a series of inequalities. Sort
of like big-O notation.
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
But then the final contradiction of the kind f(t) < f(t) (for some f)
poses no problem. If I am allowed to add some constant (not the same
on both sides of the inequality), the contradiction vanishes.
True enough. But perhaps this issue could be fixed easily by insisting
that A means A, and just changing A to A' at a few points in the paper.
This wouldn't be any sloppier than many other papers I've seen.
Your results looks fine. But the main definition of delta is not
"introduced later" - after (18), I think you point on (36) and on that
the author give attantion to (11) and (20) - but on (10) and (11). So
the main idea of the author was to show, that we need that delta to
proof RH, but there is no delta, so we can not proof RH after all (it
is not a disproof but a 'not proveable' - statement).
> Your results looks fine. But the main definition of delta is not
> "introduced later" - after (18), I think you point on (36) and on that
> the author give attantion to (11) and (20) - but on (10) and (11).
Delta is $A^*/log n$, so it is defined by A*, i.e. in (20). In (11) it
announced: "... where A* is a ... number to be further specified".
I don't mean (36) because it is deduced correctly from (20) and below.
> the main idea of the author was to show, that we need that delta to
> proof RH, but there is no delta, so we can not proof RH after all (it
> is not a disproof but a 'not proveable' - statement).
No, not so. He wants to show that if we proved RH then there would
exist such a delta (what would lead to a contradiction).
Yes, Pati by himself didn't uses the strategy showing that RH is not
proveable under the axioms of standard analysis. But indirectly - his
"disproof" shows - that the only way for standard analysis handle the
RH may lead to the conclusion, that I give before.
But you are right: He himself claims a contradicition found on RH.
My point was:
This main idea, I talked about, was unintended given by the author's
paper. But its path opened for that.
Let me see if I understand. You claim that the author wanted to show
1. RH implies the existence of delta.
2. In fact, delta does not exist.
3. Therefore, RH is false.
However, the author did not show this. But are you claiming that the
author *did* in fact show, unintentionally, that
1'. If RH were provable using standard axioms, then delta would exist.
2'. In fact, delta does not exist.
3'. Therefore, RH is not provable using standard axioms?
It would be remarkable for a paper that makes no explicit mention of
axioms or related concepts from mathematical logic to show 1'-2'-3'
without showing 1-2-3. Without looking at the paper in detail, I would
suspect that Kuznetsova's interpretation is correct, that the author
has shown neither 1 nor 1'.
So, and that is my point:
I mean that this error of Patis strategy leads us to the idea, that RH
is not proveable/ or disproveable in boundarys of standard analysis
I have read the text at the link. He says the same what I did.
Calculations are correct but it doesn't mean "Pati's paper is
If I choose a number x such that 0 < x < sin x and carry lots of
correct calculations with it -- how do you think, maybe I'll get
even more striking contradiction?
> I mean that this error of Patis strategy leads us to the idea, that RH
> is not proveable/ or disproveable in boundarys of standard analysis
It has nothing to do with axioms. It's a mere logical error of one
definite mathematician. This happens from time to time... Erratum