Looking at the Re=-1 line of the zeta function instead of the Re=-0.5 line

This looks like a good idea, because it gives the integers 1/n
weighting instead of 1/sqrt(n).

-Mike

I agree. -Carl

At 02:52 PM 9/3/2011, you wrote:
>This looks like a good idea, because it gives the integers 1/n
>weighting instead of 1/sqrt(n).
>
>-Mike
>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> This looks like a good idea, because it gives the integers 1/n
> weighting instead of 1/sqrt(n).

Yeah, I've been thinking of discussing it. It's the boundary between where zeta acts like an error measure and where it's a goodness measure.

On Sat, Sep 3, 2011 at 7:11 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > This looks like a good idea, because it gives the integers 1/n
> > weighting instead of 1/sqrt(n).
>
> Yeah, I've been thinking of discussing it. It's the boundary between where zeta acts like an error measure and where it's a goodness measure.

I don't understand why it wouldn't be an error measure either way,
with the error weighting just changing.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Sep 3, 2011 at 7:11 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > This looks like a good idea, because it gives the integers 1/n
> > > weighting instead of 1/sqrt(n).
> >
> > Yeah, I've been thinking of discussing it. It's the boundary between where zeta acts like an error measure and where it's a goodness measure.
>
> I don't understand why it wouldn't be an error measure either way,
> with the error weighting just changing.
>
> -Mike
>
So this is goodness as opposed to badness, right? I read the wiki article on zeta tuning. Do you mean Re = -1 which is the mirror of Re = 2 across the critical strip (which gives Apery's Constant pi^2/6) and of course -1 corresponding to Gigma_1,oo = -1/12....Sorry to come at this sideways, but could you bring me up to speed on why you are outside the critical strip here for zeta tuning at [-1, -0 ). I know zeta tuning is NOT along the critical line...(Do I need to reread the wiki article?) Thanks much. pgh.

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Sep 3, 2011 at 7:11 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > This looks like a good idea, because it gives the integers 1/n
> > > weighting instead of 1/sqrt(n).
> >
> > Yeah, I've been thinking of discussing it. It's the boundary between where zeta acts like an error measure and where it's a goodness measure.
>
> I don't understand why it wouldn't be an error measure either way,
> with the error weighting just changing.

My point is that if s>1, then zeta(s)-|zeta(s+i*it)| gets arbitrarily small but not zero for t>0. It becomes very small when a weighted error is very small, and in general looks like an error measure to me. When s=1, you can no longer do this, and |zeta(1/2+i*t)| seems clearly to act like a goodness measure. So s=1 seems to me to be the only place you could reasonably call the boundray.
>
> -Mike
>

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > On Sat, Sep 3, 2011 at 7:11 PM, genewardsmith
> > <genewardsmith@> wrote:
> > >
> > > --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > > >
> > > > This looks like a good idea, because it gives the integers 1/n
> > > > weighting instead of 1/sqrt(n).
> > >
> > > Yeah, I've been thinking of discussing it. It's the boundary between where zeta acts like an error measure and where it's a goodness measure.
> >
> > I don't understand why it wouldn't be an error measure either way,
> > with the error weighting just changing.
> >
> > -Mike
> >
> So this is goodness as opposed to badness, right? I read the wiki article on zeta tuning. Do you mean Re = -1 which is the mirror of Re = 2 across the critical strip (which gives Apery's Constant pi^2/6) and of course -1 corresponding to Gigma_1,oo = -1/12....Sorry to come at this sideways, but could you bring me up to speed on why you are outside the critical strip here for zeta tuning at [-1, -0 ). I know zeta tuning is NOT along the critical line...(Do I need to reread the wiki article?) Thanks much. pgh.
>

Oops. I see it got cut off, you mean 1 instead of 1/2 (why the negative signs?)

pgh

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:

> So this is goodness as opposed to badness, right? I read the wiki article on zeta tuning. Do you mean Re = -1 which is the mirror of Re = 2 across the critical strip (which gives Apery's Constant pi^2/6) and of course -1 corresponding to Gigma_1,oo = -1/12....Sorry to come at this sideways, but could you bring me up to speed on why you are outside the critical strip here for zeta tuning at [-1, -0 ). I know zeta tuning is NOT along the critical line...(Do I need to reread the wiki article?) Thanks much. pgh.

Mike was talking about Re=0, but R=-1 is a whole other deal, and Apery's Constant, by the way, is zeta(3) and comes up if you're looking at Re=3.

On Mon, Sep 5, 2011 at 4:05 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
>
> My point is that if s>1, then zeta(s)-|zeta(s+i*it)|

Should the term inside the absolute value be zeta(s+i*t)? It says
i*it, which is just -t.

> gets arbitrarily small but not zero for t>0. It becomes very small when a weighted error is very small, and in general looks like an error measure to me. When s=1, you can no longer do this, and |zeta(1/2+i*t)| seems clearly to act like a goodness measure. So s=1 seems to me to be the only place you could reasonably call the boundray.

I was actually talking about -1, not +1. But the fact that +1 behaves
that way seems interesting as well. If it's not clear why I'm saying
that -1 weights the integers at 1/n instead of 1/sqrt(n) I'd be happy
to lay it out more explicitly. I approached my analysis of the zeta
function from perhaps an unconventional standpoint, but I more or less
just assume that you know every implication and property of it that
exists, except for maybe the truth of the Riemann hypothesis, although
I bet you probably know that too but just don't realize that's it.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Sep 5, 2011 at 4:05 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> >
> > My point is that if s>1, then zeta(s)-|zeta(s+i*it)|
>
> Should the term inside the absolute value be zeta(s+i*t)? It says
> i*it, which is just -t.
>
> > gets arbitrarily small but not zero for t>0. It becomes very small when a weighted error is very small, and in general looks like an error measure to me. When s=1, you can no longer do this, and |zeta(1/2+i*t)| seems clearly to act like a goodness measure. So s=1 seems to me to be the only place you could reasonably call the boundray.
>
> I was actually talking about -1, not +1. But the fact that +1 behaves
> that way seems interesting as well. If it's not clear why I'm saying
> that -1 weights the integers at 1/n instead of 1/sqrt(n) I'd be happy
> to lay it out more explicitly. I approached my analysis of the zeta
> function from perhaps an unconventional standpoint, but I more or less
> just assume that you know every implication and property of it that
> exists, except for maybe the truth of the Riemann hypothesis, although
> I bet you probably know that too but just don't realize that's it.
>
> -Mike
>

If Gene proves RZH I wanna be in the fan-club! (a priori).

MIke, why are you working with zeta(-1)? That behaves completely differently from
+1, zeta(-1) = -1/12, that is the Sigma(1,oo) or 1 + 2 + 3 + 4 + 5 ... = -1/12 in the
analytical continuation to the complex half plane ala alternating Dirichlet series.

And zeta(1) is important because it was proven that no roots lie there...

- Paul

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
>
> > So this is goodness as opposed to badness, right? I read the wiki article on zeta tuning. Do you mean Re = -1 which is the mirror of Re = 2 across the critical strip (which gives Apery's Constant pi^2/6) and of course -1 corresponding to Gigma_1,oo = -1/12....Sorry to come at this sideways, but could you bring me up to speed on why you are outside the critical strip here for zeta tuning at [-1, -0 ). I know zeta tuning is NOT along the critical line...(Do I need to reread the wiki article?) Thanks much. pgh.
>
> Mike was talking about Re=0, but R=-1 is a whole other deal, and Apery's Constant, by the way, is zeta(3) and comes up if you're looking at Re=3.
>

No, I think he means Re = -0.5 (5 got cut off...) (Am I right?). Sorry my bad about Apery's
Constant, I was thinking Apery's Constant was Zeta(2) which is pi^2/6 (But I did say that much, pi^2/6). My point was that -1 is diametrically opposed to +2 across the critical
strip [0,1]. I read the Derbyshire and Sabbagh books so I should know better. I'll try to be more careful....(Also read the Edwards book, part of it, okay, I know, I'm still not ready for that one, except I reviewed Schaum's Complex Analysis and then retackled it)

Per Terry Gannon there is some significance to this, and Z/48 and moonshine of all things...sorry I know my posts are clusters, I don't think I will ever improve that way...

So, is all this just a novel usage of the Riemann Zeta Function, or do tuning systems with
"goodness" really have any bearing on the truth of RZH? (Or vice versa, is it just as simple
as very good systems on 1/2 + it?) Is that the whole thing about the Gram points..

PGH

On Mon, Sep 5, 2011 at 11:12 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> I was actually talking about -1, not +1.

By the great white beard of Zeus! I'm talking about +1 after all.

I'll have to check out values that are > 1 to see just how goodnessy
things can get.

-Mike

On Tue, Sep 6, 2011 at 1:09 AM, Paul <phjelmstad@msn.com> wrote:
>
> MIke, why are you working with zeta(-1)? That behaves completely differently from
> +1, zeta(-1) = -1/12, that is the Sigma(1,oo) or 1 + 2 + 3 + 4 + 5 ... = -1/12 in the
> analytical continuation to the complex half plane ala alternating Dirichlet series.

By Neptune's watery trident! I did in fact mean zeta(1).

> And zeta(1) is important because it was proven that no roots lie there...

Yeah, that was another thing I was telling Paul - there shouldn't be
any temperaments that really have a utility of zero, right?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Sep 5, 2011 at 11:12 PM, Mike Battaglia <battaglia01@...> wrote:
> >
> > I was actually talking about -1, not +1.
>
> By the great white beard of Zeus! I'm talking about +1 after all.
>
> I'll have to check out values that are > 1 to see just how goodnessy
> things can get.
>
> -Mike
>

Somehow I thought you were! Even if -1 made sense, -0.5 just doesn't :)

On Tue, Sep 6, 2011 at 1:32 AM, Paul <phjelmstad@msn.com> wrote:
>
> Which Paul? Well, unless you can afford a staircase that leads nowhere, just for show,
> (good grief....)

I'm talking about Paul Erlich; we worked on it while I was up at his
place in Boston a little while ago.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Sep 6, 2011 at 1:32 AM, Paul <phjelmstad@...> wrote:
> >
> > Which Paul? Well, unless you can afford a staircase that leads nowhere, just for show,
> > (good grief....)
>
> I'm talking about Paul Erlich; we worked on it while I was up at his
> place in Boston a little while ago.
>
> -Mike
>

Well, since all the roots are on 1/2 + it, (or so it would seem...) shouldn't
all the focus be there? Or, near it anyway.

of course, with t = 2 pi n / ln 2 in z = x +it apparently high values along the line x > 1 are best. 1 is a pole, and then of course > 1 they all converge...so I guess this is the best place to be if you are not on a the line 1/2 where all the roots purportedly are, that is non-trivial and I mean, well -2 -4 -6 etc are RIGHT OUT! (are they?) the trivial ones...all I remember now is that there is symmetry across the line 1/2 but also across the x axis, but I don't know if flipping the imaginary part really matters.

pgh.

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Sep 5, 2011 at 4:05 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> >
> > My point is that if s>1, then zeta(s)-|zeta(s+i*it)|
>
> Should the term inside the absolute value be zeta(s+i*t)? It says
> i*it, which is just -t.

Of course.

> I was actually talking about -1, not +1.

Why?