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convexity conjecture

Synonym: 
Hardy-Littlewood convexity conjecture
Type of Math Object: 
Conjecture
Major Section: 
Reference

Mathematics Subject Classification

11A41 no label found

Comments

Why is this called "convexity conjecture"? I've read the reference, too, and they don't explain it at all.

> Why is this called "convexity conjecture"? I've read the
> reference, too, and they don't explain it at all.

Here's a first approximation: if $\pi$ were a convex function,
then it would follow that
\[
\pi( (x+y)/2 ) \le (\pi(x) + \pi(y)) / 2.
\]
But \pi( (x+y)/2 ) doesn't make sense if $x+y$ is odd, so
let's drop the 2s:
\[
\pi(x + y) \le \pi(x) + \pi(y).
\]
I'm a bit uneasy about this, since $\pi(x/2)$ is not the same
as $\pi(x)/2$, but that may also be the reason Crandall and
Pomerance put ``convexity'' in scare quotes.

I hope an analyst will pop in to clear things up.

Do they say who originally came up with this conjecture.
Presumably, to find out why the term "convexity conjecture",
you need to know who came up with the term and then see how
that person justified the term. If that person didn't say
anything, then Michael Slone's guess seems as good as any.

OOps, you already do know who came up with it. So then the
question becomes to to figure out in where Hardy and Littlewood
first stated this conjecture, read the reference, and see what
they had to say. Of course, it may happen that they weren't
the ones who termed it "convexity conjecture", in which case
you might need to spend time in the library tracking down forward
references to their original article to figure out who exactly
came up with the term and why. At that point, it becomes an
issue of whether the question is interesting enough to bother
getting to the bottom ofthe issue.

Think easier, think progressive.

Hardy and Littlewood lived in simpler times, when mathematicians weren't as concerned as they are today with piling up abstractions and generalizations like floors on a house of cards.

They were able to visualize things like the Julia fractals. So what if we take PrimeFan's n and change it to z (an axis)? It then becomes very easy to visualize what kind of shape would result if this conjecture was to be proven false and we plotted at the coordinates given by the counterexample.

But don't take my word for it, put it through your favorite CAS. In Mathematica, you can play around with

Plot3D[PrimePi[x + y] - (PrimePi[x] + PrimePi[y]), {x, 1, 5000}, {y, 1, 5000}]

or

Plot3D[PrimePi[x] + PrimePi[y], {x, -2500, 2500}, {y, -2500, 2500}]

etc.

I don't think anyone here would take your word for it if you said 1 plus 1 is 2 (giggle). But seriously, you make a good point about the helpfulness of plotting tools. Per Cran & Pom, I'm gonna try taking these plots to x = y = 20000 if my computer can handle it.

> At that point, it becomes an
issue of whether the question is interesting enough to bother
getting to the bottom ofthe issue.

For me it is interesting enough, plus it gives me another reason to go down to the library (when I just have one thing to look up, it's just not a priority with everything else I have going -- now I have this plus the Lucas-Lehmer code and a couple of topics Bob Happ asked me about.

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