computing the Riemann zeros

[RichD]

I was reading a piece on the Rieman hypothesis,
and that 300 million zeta function zeros have been
computed.

How the heck do they do the math, numerically?
They're using digital logic, with finite word length,
yes/no? That means roundoff errors. The hypothesis stipulates the
zeros EXACTLY, not to 32 bits precision.

--
Rich

[RichD]

I was reading a piece on the Rieman hypothesis,
and that 300 million zeta function zeros have been
computed.

How the heck do they do the math, numerically? They're using digital
logic, with finite word length, yes/no? That means roundoff errors.
The hypothesis stipulates the zeros EXACTLY

--
Rich

[micro...@hotmail.com]

Zero is just a name for the abscence of a quantity.
It is the only number...all others are quantities with names.
It is needed to define bases. That is the function of zero.

Mitchell Raemsch; the prize

[Tonico]

Well, no: the hypothesis only says what the real part of the non-
trivial zeros is, it stipulates nothing else.

Tonio

[Tonico]

On Feb 16, 5:48 am, "microm2...@hotmail.com" <microm2...@hotmail.com>
wrote:

Idiot

[Pfs...@aol.com]

So what has this to do with the question!

[David C. Ullrich]

On Wed, 15 Feb 2012 17:52:08 -0800 (PST), RichD
<r_dela...@yahoo.com> wrote:

>I was reading a piece on the Rieman hypothesis,
>and that 300 million zeta function zeros have been
>computed.

Is that what it said, or did it just say that the RH had been
verified for the first 300 million zeroes? It makes a big
difference!

I don't know how people actually do these computations.
But I can explain how at least theoretically it's possible
to know that a zero of the zeta function lies _exactly_
on the line s = 1/2, in spite of roundoff errors in
the calculations:

1. Say f is an analytic function and c is a closed curve.
You can verify numerically that f has no zero _on_ c
be first getting a bound of the derivative of f, then
calculating f at a finite number of points to suffiicient
precision.

A simpler example of how that might go: Suppose
you have f defined on [0,1]. Suppose you know
that |f'(t)| <= C everywhere, where C is _approximately_ 1.
Suppose you know that f(0) is _approximately_ 5.
Then (assuming those approximations are good enough)
you know that f has no zero on [0,1], because you
know that |f(t) - f(0)| >= |f(0)| - Ct >= 4, approximately.

2. Magic: Given an analytic function f and a closed
curve c, such that f has no zero on c (which you can
know in spite of roundoff error - see (1) above),
there is a certain integral that gives exatly the
number of zeroes of f inside c. So you can calculate
the number of zeroes inside c exactly in spite of
roundoff error, because the number is an integer;
if there are 2 zeroes inside c, with an error less than 1/2,
then there are exactly 2 zeroes inside c.

3. More magic: The zeta function satisfies certain
identites, showing that if there was a zero in
the critical strip but not on the critical line, then
there would be another zero nearby. SO: You
calculate that there is exactly one zero inside
a certain tiny circle (see (2)) and it follows that that
zero is _exactly_ on the line s = 1/2.

[Tim Norfolk]

Years ago, te Riele found something like 1.5 billion zeros. What they
did was to find sign changes on the line Re z = 1/2, and show that
there were no other zeros, not find the exact values of the zeros
themselves.

[Frederick Williams]

"micro...@hotmail.com" wrote:

>
> Zero is just a name for the abscence of a quantity.
> It is the only number...all others are quantities with names.
> It is needed to define bases. That is the function of zero.

I cannot begin to tell you how much I valued your contribution.

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

[Pubkeybreaker]

On Feb 16, 10:17 am, David C. Ullrich <ullr...@math.okstate.edu>
wrote:

> On Wed, 15 Feb 2012 17:52:08 -0800 (PST), RichD

> 3. More magic: The zeta function satisfies certain
> identites, showing that if there was a zero in
> the critical strip but not on the critical line, then
> there would be another zero nearby.

Yes. This is at the heart of the question.

If there is a zero OFF the critical line
it will have a matching zero on the other side of the critical line.

(if there is a zero at (a+bi), there will be another at (1-a) + bi)

[Pubkeybreaker]

No, it does NOT. It stipulates the REAL PART exactly.

[RichD]

On Feb 16,, Pubkeybreaker <pubkeybrea...@aol.com> wrote:
> > I was reading a piece on the Rieman hypothesis,
> > and that 300 million zeta function zeros have been
> > computed.
>
> > How the heck do they do the math, numerically?  They're using digital
> > logic, with finite word length, yes/no?  That means roundoff errors.
> > The hypothesis stipulates the zeros EXACTLY
>
> No, it does NOT.  It stipulates the REAL PART exactly.

Yes, it DOES.
it stipulates both exactly.

It does not claim "the zeros of the zeta function, +- 2^-32
precision, all lie on the line of real part = 1/2 +- 2^-32",
as far as I know (mein deutsch is a bit weak, admittedly).

--
Rich

[Tonico]

Either you don't understand what the Riemann's Hypothesis exactly is/
says or else you don't know the verb "stipulate" = specification or
demand of conditions or STIPULATIONS in a contract, agreement or
pact".

Well, RH ONLY states that the non-trivial zeros of the (Euler-Riemann)
zeta function have real part equal to 1/2. Period. It says NOTHING
about their imaginary part.

So yes: as "far as you know" you're wrong.

Tonio