
A possible proof for Riemann Hypothesis - nutanc
I work with a professor,Dr. Kumar Eswaran, on several computer science problems especially in the neural network field. He is the quintessential prof who is always lost in thought and interested in solving unsolvable problems. So I was pleasantly surprised last week when he told me that he had found a proof for the Riemann Hypothesis.<p>The Professor was able to crack the Riemann Hypothesis because he used the properties of primes and analytic continuation and had a new way of handling slowly converging series and was able to use (at crucial point) concepts borrowed from Donald Knuth regarding random numbers and random sequences. Knuth had said that for any sequence to be truly random it has to be non-cyclic. The proof required to show that a sequence of +1&#x27;s and -1&#x27;s , obtained from the prime factorization of the infinite sequence of integers, had to be shown to be random and to  asymptotically behave like the tosses of a coin.<p>Though I am good at computers and math, I am not an authority in math. So I thought I will put up the proof here and invite comments as Dr.Kumar does not know too much about Hackernews.<p>Links to proof:<p>https:&#x2F;&#x2F;arxiv.org&#x2F;pdf&#x2F;1609.06971v4.pdf<p>https:&#x2F;&#x2F;www.researchgate.net&#x2F;publication&#x2F;309205618_The_Dirichlet_Series_for_the_Liouville_Function_and_the_Riemann_Hypothesis
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mathematician23
The function he is trying to prove is analytic is a ratio of the zeta function
at two places, so can be thought of as a ratio of two different complex
functions. What if they both have a zero for some complex number s, i.e.
zeta(2s)/zeta(s) = 0/0, and l'hospitals rule ends up showing its a removable
singularity and so the ratio is still analytic, like he says. In that case, he
hasn't shown there are no zeros in the critical strip outside the critical
line.

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sainik
This does not happen. The zeros on the critical line appear as poles in F(s).
There are no other poles,in F(s), because The trivial zeros of the zeta
function which occur at -2,-4,-6 etc actually cancel out from the numerator
and the denominator (i.e. from zeta(2s) and from zeta(s)). The paper then
shows that the only poles in F(s) are those on the critical line, thus proving
R.H.

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mathematician23
There are many equivalent statements to the Riemann Hypothesis (RH) and
statements that, if true, would imply the RH holds. But F(s) =
zeta(2s)/zeta(s) being analytic in the critical strip, outside the critical
line, DOES NOT imply the RH is true.

That's my point, it doesn't matter if the author really showed F(s) is
analytic in that region or not.

Showing something is equivalent to the RH is a major area of research and so
is finding statements that directly imply the RH. The function F(s) is very
well known and a simple function, so if there were any simple statements
involving it that directly implies the RH, we would know by now. I hope this
helps.

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vskp2016
The paper clearly demonstrates that if F(s) has poles on the critical line, RH
is proved.This is because the poles in F(s) correspond to zeros of zeta(s) and
it is well known that there are no other zeros outside the critical strip
(except for the trivial zeros). The paper has proved that the only poles of
F(s),occur on this critical line. Ok bye!

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cokernel
It's not a good sign that it's classified as math.GM. One would normally
expect a real proof to be classified more finely. To be fair, the lack of
proper classification does not strictly imply that it's not a real proof.

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noobermin
Not a mathematician, but I'd wait to see what refree report/editors say(s). If
this is validated, this is huge.

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deepnotderp
Wow, this is super interesting. I'm gonna wait on results. Btw,love that you
published it in arxiv.

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nutanc
Yes, arxiv for the win :)

