Yahoo Tuning Groups Ultimate Backup tuning-math Convolution Theorem erratum and the way forward for all this

I discovered an error in my current formulation of the theorem, albeit
one that makes the whole thing a lot easier to solve.

http://www.mikebattagliamusic.com/music/HEConvolutionTheorem.html

Note step #9: I factor out the Gaussian term because of the property
that convolution associates with scalar multiplication. The G(x) term
isn't scalar, though, so this doesn't apply. This actually makes
things simpler, because it now turns out that my intuitions in
representing the basis function with a bunch of impulses were right
on.

So to fix this, I need to figure out the following, which I don't understand:

1) What is the actual width of each interval? Is it really
1/sqrt(n*d)? Paul said he determined this empirically - is there any
proof for this?
2) It isn't 1/sqrt(n*d) for all series, correct? The Farey series had
width 1/d, I believe. So does this mean that HE basically works only
because these series are set up the way that they are? I understand
some kind of "unimodular" property is involved, but does this really
reflect something important about the distribution of the rationals,
or just the series we choose to use?
3) In general, as far as the rationals are concerned, does the
"relative density" of the rationals actually vary across the line? The
rationals are completely dense, but in an infinite sense, are there
actually "less" of them near 3/2 than near 16/15? Is it that the
density of the rationals follows a sqrt(n*d) pattern?

If #3 isn't true, and it turns out that as the full set of rationals
is added the relative density is uniform, then that's bad news
mathematically, since it means that Paul's original model only works
because a finite subset of the rationals is being used. At this point
I'd have to stop trying to "prove" things and start trying to "design"
things to just make the model do what I want.

But, if I can work this out, then I think it will also mean that it's
now possible to compute HE in insanely fast time by using the
undersampling trick from before. As of right now, though, the proof as
it stands is broken, but apparently still close enough to produce a
decently workable model. This is probably what was causing all of the
bugs in my algorithm.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> 3) In general, as far as the rationals are concerned, does the
> "relative density" of the rationals actually vary across the line?

That's partly what ?(x) is all about--it has a derivative of zero at rational numbers, recall. How regular the Farey sequence is is not known, since a statement about it is equivalent to the Riemann hypothesis: the sum of the difference in absolute value between the ith Farey number of order n and i/m, where m is the number of fractions of order n, is O(n^(1/2+e)), e any e>0.

🔗Mike Battaglia <battaglia01@gmail.com>

On Mon, Jan 24, 2011 at 6:25 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> http://www.mikebattagliamusic.com/music/HEConvolutionTheorem.html
>
> Note step #9: I factor out the Gaussian term because of the property
> that convolution associates with scalar multiplication. The G(x) term
> isn't scalar, though, so this doesn't apply. This actually makes
> things simpler, because it now turns out that my intuitions in
> representing the basis function with a bunch of impulses were right
> on.

Note: I spent lots of hours last night fixing the above, so just
ignore this now. I'm still not sure it's entirely rigorous, because of
the approximation that I used, but I think it is. I need to have
someone who's a lot more familiar with what rigor is necessary when it
comes to statements regarded the limit of the behavior of the limit of
a finite series approaching an infinite one.

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Mon, Jan 24, 2011 at 8:31 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > 3) In general, as far as the rationals are concerned, does the
> > "relative density" of the rationals actually vary across the line?
>
> That's partly what ?(x) is all about--it has a derivative of zero at rational numbers, recall. How regular the Farey sequence is is not known, since a statement about it is equivalent to the Riemann hypothesis: the sum of the difference in absolute value between the ith Farey number of order n and i/m, where m is the number of fractions of order n, is O(n^(1/2+e)), e any e>0.

This is awesome, I'm finally starting to understand this stuff. Maybe
we'll actually solve RH on this list, haha. It continues to frustrate
the hell out of me that I can't just wrap my head around everything
and figure out exactly what the Riemann hypothesis IS, though, or what
it's actually saying. I'll respond to these one by one:

> That's partly what ?(x) is all about--it has a derivative of zero at rational numbers, recall.

I need to read up on ?(x), but - does d?(x)/dx only vanish at the
rational numbers? I don't understand exactly how this is related to
the density of the rationals... can you explain? Is ?(x) actually
continuous?

If there's a derivative of zero at the rational numbers, then it
doesn't seem like it would yield a good answer to the question...

> How regular the Farey sequence is is not known, since a statement about it is equivalent to the Riemann hypothesis

What do you mean by "regular?" As in, how well the Farey sequence
approximates the actual distribution of the rationals? Does the same
apply to the Stern-Brocot tree and such? I guess it's because the
distribution of the rationals is related to the distribution of the
primes, which is subject to the Riemann hypothesis?

Is it just that the actual distribution of the rationals is as poorly
understood as the actual distribution of the primes?

> the sum of the difference in absolute value between the ith Farey number of order n and i/m, where m is the number of fractions of order n, is O(n^(1/2+e)), e any e>0.

What do you mean the "sum of the difference in absolute value"
here...? Let F_n,i be the ith Farey number of order n. You're saying
that

Sum_i |F_n,i - i/m| = O(n^2(1/2+e))

Thanks for the help here.

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
It continues to frustrate
> the hell out of me that I can't just wrap my head around everything
> and figure out exactly what the Riemann hypothesis IS, though, or what
> it's actually saying. I'll respond to these one by one:

There are a number of equivalent formulations which look entirely different.

> I need to read up on ?(x), but - does d?(x)/dx only vanish at the
> rational numbers?

Yes. It's continuous, with a zero derivative at rational numbers, and an undefined one elsewhere.

> If there's a derivative of zero at the rational numbers, then it
> doesn't seem like it would yield a good answer to the question...

I haven't been able to figure out what the question is.

> > How regular the Farey sequence is is not known, since a statement about it is equivalent to the Riemann hypothesis
>
> What do you mean by "regular?" As in, how well the Farey sequence
> approximates the actual distribution of the rationals?

How close does it come to being equally distributed in the unit interval.

> Sum_i |F_n,i - i/m| = O(n^2(1/2+e))

Right; a similar claim for squared difference.

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Tue, Jan 25, 2011 at 10:23 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> It continues to frustrate
> > the hell out of me that I can't just wrap my head around everything
> > and figure out exactly what the Riemann hypothesis IS, though, or what
> > it's actually saying. I'll respond to these one by one:
>
> There are a number of equivalent formulations which look entirely different.

I get the gist of it is that the real component of the Mellin
transform corresponds to some kind of damping factor, so if the zeros
are all on the critical line then the damping rate won't overtake the
rate of Li(x), meaning that Li(x) is a pretty good estimate for pi(x)
for all of infinity, yes? Am I on the right track?

> > If there's a derivative of zero at the rational numbers, then it
> > doesn't seem like it would yield a good answer to the question...
>
> I haven't been able to figure out what the question is.

Specifically I mean question 3 from my first post:

In general, as far as the rationals are concerned, does the
"relative density" of the rationals actually vary across the line? The
rationals are completely dense, but in an infinite sense, are there
actually "less" of them near 3/2 than near 16/15? Is it that the
density of the rationals follows a sqrt(n*d) pattern?

Paul and I have argued about this for a long time. I think that the
sqrt(n*d) widths are due to the Tenney series, and that the rationals
are evenly distributed. He thinks there really are more rationals near
100001/100000 than near 2/1.

I don't understand how the answer can be both yes and no.

> > > How regular the Farey sequence is is not known, since a statement about it is equivalent to the Riemann hypothesis
> >
> > What do you mean by "regular?" As in, how well the Farey sequence
> > approximates the actual distribution of the rationals?
>
> How close does it come to being equally distributed in the unit interval.

How close do the actual rationals themselves come to being equally
distributed in the unit interval?

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > You know, you could always do a little convolving on ?(x) and get a much more nicely behaved function.
> >
>
> I'd be interested in seeing what you get by convolving ?(x) with various normal distributions and taking the derivative.
>

Here's an article on the Weierstrass transform which may be useful. Note particularly what it says about the Weierstrass transform being holomorphic; the transform of ?(x) will be holomorphic and you can happily take its derivative.

http://en.wikipedia.org/wiki/Weierstrass_transform

🔗Mike Battaglia <battaglia01@gmail.com>

On Tue, Jan 25, 2011 at 10:57 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
>
> Here's an article on the Weierstrass transform which may be useful. Note particularly what it says about the Weierstrass transform being holomorphic; the transform of ?(x) will be holomorphic and you can happily take its derivative.
>
> http://en.wikipedia.org/wiki/Weierstrass_transform

I saw this a while ago - I wasn't sure what the point of giving
convolution with a Gaussian its own super official "transform" name
was, but hey, why not? :)

So then, to restate my hypothesis in one sentence, Harmonic Entropy is
the Weierstrass transform of this function:

http://en.wikipedia.org/wiki/Thomae%27s_function

iff a Farey series is used.

And since Thomae's function is a projective view of Euclid's orchard,
maybe you can see now what I was talking about when I said we'd be
able to tie this into regular mapping if it were true.

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Tue, Jan 25, 2011 at 10:34 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> > You know, you could always do a little convolving on ?(x) and get a much more nicely behaved function.
> >
>
> I'd be interested in seeing what you get by convolving ?(x) with various normal distributions and taking the derivative.

The same thing that would happen if you first took the derivative of
?(x) and then convolved, or convolved the derivative of a Gaussian
with ?(x). I'll give it a go.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I get the gist of it is that the real component of the Mellin
> transform corresponds to some kind of damping factor, so if the zeros
> are all on the critical line then the damping rate won't overtake the
> rate of Li(x), meaning that Li(x) is a pretty good estimate for pi(x)
> for all of infinity, yes? Am I on the right track?

The prime number theorem is proven just by reference to zeta along the line s=1, not s=1/2; it has a simple pole at zero and no other singularities, and this is the data you need, together with the fact that the zeros in the critical strip are bounded away from s=1. The Riemann hypothesis allows you to improve the error bound in the prime number theorem, because the zeta zero oscillation terms of psi(x) will be less.

> Paul and I have argued about this for a long time. I think that the
> sqrt(n*d) widths are due to the Tenney series, and that the rationals
> are evenly distributed. He thinks there really are more rationals near
> 100001/100000 than near 2/1.

Depends on which measure you pick. In the ordinary dx measure, they are even.

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Jan 25, 2011 at 10:34 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> > > You know, you could always do a little convolving on ?(x) and get a much more nicely behaved function.
> > >
> >
> > I'd be interested in seeing what you get by convolving ?(x) with various normal distributions and taking the derivative.
>
> The same thing that would happen if you first took the derivative of
> ?(x) and then convolved, or convolved the derivative of a Gaussian
> with ?(x). I'll give it a go.

You can't do much with ?'(x). The point is this seems to be to be pretty straightforward and should give something like HE.

🔗Mike Battaglia <battaglia01@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

?(x)-x is a continuous periodic function with period 1, and hence has a Fourier series expression. It's trivial to find the Fourier series of the convolution, and hence of the derivative of the convolution, if you have the Fourier series of ?(x)-x. Might be useful.

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Jan 26, 2011 at 12:28 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> ?(x)-x is a continuous periodic function with period 1, and hence has a Fourier series expression. It's trivial to find the Fourier series of the convolution, and hence of the derivative of the convolution, if you have the Fourier series of ?(x)-x. Might be useful.

This function is giving me a headache. I just spent some time on it,
but the one MATLAB implementation I could find was flawed. Let's come
back to this, because I think I've found some nifty speedups
elsewhere.

-Mike