
The De Bruijn-Newman constant is non-negative - lisper
https://terrytao.wordpress.com/2018/01/19/the-de-bruijn-newman-constant-is-non-negativ/
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vincentchu
A bit rusty on the math, but taking a quick gander at the Wikipedia page
([https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_const...](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_constant)):

"In brief, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0."

This result seems to imply that the De Bruijn-Newman constant (Λ) is non-
negative, i.e., >=0. Thus, were one able to prove that Λ=0, then one would
have also shown the Riemann Conjecture to be true? Similarly, if one were to
show Λ to be not equal to 0, then the Riemann Conjecture would be false.

Again, just a quick reading. Sure others who are better/smarter can chime in!

Update: Tao expands on this point in the bottom of his blog post.

~~~
CogitoCogito
Yes so if you want to prove the Riemann hypothesis one possibility would be to
similarly show some sort of contradiction if Λ > 0\. This is sort of the
opposite approach that Rogers and Tao have appeared to use. Of course maybe
that side is much much harder (it is equivalent to the Riemann hypothesis
after all).

~~~
vincentchu
I have essentially no real knowledge about this topic, but I suppose some
progress has been made by tightening the bounds for this constant?

Can somebody with more info chime in--- is this a huge, groundbreaking amount
of progress?

~~~
CogitoCogito
I would presume that entirely different methods would be needed to prove that
the constant can't be positive. There are many types analytical proofs in math
that break down to "negative", "zero" and "positive" cases and often a couple
of those are relatively easy whereas others are extremely difficult. The
Calabi conjecture is an example.

So really it's hard to say. You might even say this proof made things harder.
It guarantees that you have to prove the constant is 0 if you want to prove
the conjecture. Before you might have hoped prove negativity.

~~~
cookingrobot
Right, near the bottom of the paper he writes “...this result does not make
the Riemann hypothesis any easier to prove, in fact it confirms the delicate
nature of that hypothesis”

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ajkjk
I wonder if it bothers someone like Terry Tao that a question of this requires
_so much machinery_ to solve. I'd like to believe him that, say, the work to
renormalize the Hamiltonian there is totally correct and rigorous -- he's the
best person to believe that it is, anyway. But why is it... necessary?

It seems so completely unlikely that the universe would require going so far
afield from the original question in order to prove something that's
relatively simple to state about it!

Of course, it's probably because there's some big piece missing -- if you knew
how to prove the RH, this would probably be a comparatively trivial corollary.
But it _still_ feels kinda weird that all those conclusions couldn't be
'reduced' to something simpler.

~~~
danharaj
Undecidability basically guarantees the existence of arbitrarily hard to prove
but easy to state theorems.

~~~
posterboy
Why is the completeness theorem never mentioned in this context? FOL is
decidable! And as far as I can see, RH is in FOL.

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fspeech
Scroll to the end of his blog entry. There Terry Tao gives an addendum
explaining in layman's terms the meaning of their work:

"ADDED LATER: the following analogy (involving functions with just two zeroes,
rather than an infinite number of zeroes) may help clarify the relation
between this result and the Riemann hypothesis (and in particular why this
result does not make the Riemann hypothesis any easier to prove, in fact it
confirms the delicate nature of that hypothesis). ..."

Probably should have been placed at the beginning of the blog for the main
content is certainly intimidating.

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piemonkey
Can someone with more knowledge than me contextualize this result? It appears
to be making progress on the Riemann hypothesis, but why is this particular
effort getting attention on HN?

~~~
feniv
For context, the author is the notable fields medal winner Terence Tao -
[https://en.wikipedia.org/wiki/Terence_Tao](https://en.wikipedia.org/wiki/Terence_Tao)
\- and finding the lower bounds for this constant has been a long running
endeavor in pure mathematics -
[https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_const...](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_constant)

~~~
cpfohl
Most useful comment here.

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kleiba
_If we normalize the Fourier transform {{\mathcal F} f(\xi)} of a (Schwartz)
function {f(x)} as {{\mathcal F} f(\xi) := \int_{\bf R} f(x) e^{-2\pi i \xi
x}\ dx}, it is well known that the Gaussian {x \mapsto e^{-\pi x^2}} is its
own Fourier transform._

Funny that he should mention it: just the other day, my mom and I were having
a good laugh because as it turned out mom's neighbor, a Mrs. Cringleton,
apparently had no idea that the Gaussian is its own Fourier transform under
these circumstances. Apparently it came up over a cup of tea and bicuits they
were having, and Mrs. Cringleton felt quite a bit embarrassed when my mom
remarked: "But my dear, that is well-known!"

Poor Mrs. Cringleton...

~~~
dxbydt
Back when I worked at Twitter, celebs used to show up randomly at the HQ at
lunchtime. I was just your average Indian immigrant data scientist, not clued
into American culture.

The team I was on was trying to hire a lady for a front end eng role. I didn’t
know what the lady’s name was, but I was told she was a very promising
candidate, and we were all looking forward to the day when she joined because
we had a ton of frontend work.

Ok, so at lunch, my colleague said out aloud - Hey Katy Perry is in the
lunchroom! Everyone got excited. So I said, Oh is that the new front end
engineer ? The whole room exploded in laughter. I thought I must have made a
big social gaffe. My mind quickly raced...,oh maybe she must be a celebrity of
some kind. Perry Perry...which Perry do I know...So previous night I was
watching some show on TBS and suddenly a Perry name flashed into my brain, so
I instantly said - Oh she must be the daughter of Tyler Perry!

The room just doubled up in laughter. Finally I did a wiki lookup and found
out who she was. But for days, people would look at me, laugh and say he’s the
Katy Perry Tyler Perry guy ha ha ha.

Finally I was so embarrassed and pissed off I asked them - Do you know the
difference between between Jacobian and a Hessian ? Nobody knew. Well, I said
triumphantly, I might not know your katy perry from tyler perry, but atleast I
know what I need to know to fo my work.

So yeah, some of us do know that a Gaussian is its own Fourier transform,
derivative of an exponential is the exponential itself, etc.

~~~
pierrebai
So, how does twitter uses Jacobian?

~~~
jordigh
A Jacobian is often used in optimisation. In machine learning, optimisation is
often relabelled "learning", because they're finding the minimum of a cost
function, also sometimes called an error function.

You may have heard of backpropagation. Backprop is merely a convenient way to
compute the Jacobian of a feed-forward neural network. The Jacobian computed
by backprop is then used to pick a line search direction to minimise the
network's error.

Put it another way, a Jacobian is kind of a multidimensional gradient. It
tells you in which multidimensions a function is changing the most rapidly.

~~~
dnautics
yeah but is most of twitter really using that?

~~~
dxbydt
no, but most of twitter isn't using knowledge of katy perry's whereabouts
either.

Storing garbage info like katy perry kim kardashian etc. in your brain, in my
humble opinion, is a tax on your cognition - one pays that tax by being
ignorant of elementary methods like fourier transform or multivariable
calculus or eigenvalues or trig integrals or T distribution or you get the
idea...

~~~
dnautics
I'm quite good at math and found your anecdote funny (yes, I know what a
jacobian and a hessian are, and give have lectures on using the fourier
transform to professional scientists), but honestly most people, including
coders, don't need that, and their lives are not bettered by knowing these
things. It is a lot bit pretentious, and not humble at all, to judge people
for keeping 'garbage' information.

~~~
dxbydt
I have no opinion on what others should deposit in their brains. Am just
saying personally if I start keeping track of garbage info like kate and kim,
I tend to forget my trig integrals. I have empirically verified that over
time. So given a choice, I choose to be blissfully unaware of these cultural
cues, so I can store stuff I deem more useful

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japaget
Preprint of paper in arxiv.org:
[https://arxiv.org/abs/1801.05914](https://arxiv.org/abs/1801.05914)

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daodedickinson
I'd rather not follow France on a guilt trip.

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meuk
While I thoroughly enjoy Terry Tao's work and am a big admirer of his down-to-
earth philosophy, I don't really understand why this particular article is so
popular. Indeed, the top comment (which made me LOL, btw) seems to reflect the
same attitude.

A much more accesible and enjoyable (though still somewhat technical) article
is [this one] [1]

[1]
[https://www.google.nl/amp/s/terrytao.wordpress.com/2010/04/1...](https://www.google.nl/amp/s/terrytao.wordpress.com/2010/04/10/the-
euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-
analytic-continuation/amp/)

