Yahoo Tuning Groups Ultimate Backup metatuning Has this man proven the Riemann hypothesis?

Hi Kraig,

> which is?

The Reimann zeta function is best
known for its use to predict in a
statistical fashion how
the distribution of primes
tails off as it goes to
infinity.

Then it is a complex function
- complex in the sense of a number
that can be used to solve equations
that have no real solutions.
such as x^2=-1 etc.

Probably you know this but
just to assume nothing for anyone
who is reading this:

Just as x*2=3 has no solution in
integers forcing one to create
fractions to solve such equations,
so x^2 = -1 has no solution as real
numbers but if you create complex
numbers you can solve equations
like that one too.

You find that the complex nubmers
can be separated into a real and an imaginary part
a + ib
so they are essentially two dimensional
numbers - each one specifies a position
in a plane rather than along a line.

Then the function is not only defined for
complex numbers, but is complex valued too.
So, it maps from complex numbers to complex
numbers over the surface of the
complex plane.

A real valued complex function would
just be a surface.

A complex valued complex function like
this makes a kind of twisted
sheet and strictly speaking it
is a surface in four dimensions
which you can't really visualise.

The hypothesis is about where its zeros
lie - where it takes the value 0 in
the complex plane - in other words,
where that twisted four dimensional
surface hits the 0 plane.

It has some zeroes already
well understood - and then the
hypothesis says that all its
other zeroes lie along a particular
line in the complex plane - the
line of all numbers with real
component 0.5

I.e. 0.5 + a*i where i is sqrt(-1)

Luckily it is also real valued
along that line so it is easy
to work with the cross section
of the function along that line.

There's an applet here that shows
the cross section of the Reimann surface
along the real = 0.5 line which is fun to play
with:

http://www.math.ubc.ca/~pugh/RiemannZeta/RiemannZetaLong.html

The connection with the primes is seen in the formula
at the top of that page

Its value for s can be defined as the sum of
n^-s over all n.

However with a bit of algebraic manipulation
involving expanding the ns into their
factors which I don't remember now,
you can show that that is also the
same as the product of 1/(1 - p^-s)
over all the primes p (primes this time
not integers) which is a fascinating
equivalence because the primes are
very hard to deal with in formulae
especially in a formula involving
all the primes like this one.

The way it works is that each 1/(1 - p^-s)
there can be expanded into an infinite series
and then when you take the product
of all those infinite series in a
careful way you find that they
reconstitute all those
terms in the original sum of the
n^-s. I remember doing that as
an undergraduate in my number theory
classes but can't remember
the details now.

I see here that that is one of the
results of the famous eighteenth century
Swiss mathematician Euler
http://en.wikipedia.org/wiki/Riemann_zeta_function

Then as it says on that page, you need
to know the positions of the zeroes
of the Zeta function to work out the
distribution of the prime numbers.
I remember studying that but can't remember
the details now at all.

However, it says there that there is a connection
with tuning. So I suppose that is
the main interest here. Can anyone
tell me, what is the connection
of the zeta function with tuning
anyone?

Robert

🔗Paul Erlich <PERLICH@...>

It's a little bit technical (you can easily look it up if you wish),
but it is the #1 most important unsolved mathematical problem in
existence. There is a huge edifice of mathematics whose truth is
conditional on that of the Riemann hypothesis. A proof or disproof of
the Riemann hypothesis would have a vast impact on the entire field
of number theory as well as many related areas. Though a single
counterexample hasn't come up in millions or billions of tests,
that's no guarantee one won't -- a related hypothesis was believed
true until a counterexample was found way up in the astronomical
range (I think this is is Derbyshire's book).

--- In metatuning@yahoogroups.com, kraig grady <kraiggrady@a...>
wrote:
> which is?
>
> Carl Lumma wrote:
>
> > http://www.math.purdue.edu/~branges/
> >
> > -C.
> >
> >
> >
>
> -- -Kraig Grady
> North American Embassy of Anaphoria Island
> http://www.anaphoria.com
> The Wandering Medicine Show
> KXLU 88.9 FM WED 8-9PM PST

🔗Paul Erlich <PERLICH@...>

--- In metatuning@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:

> However with a bit of algebraic manipulation
> involving expanding the ns into their
> factors which I don't remember now,
> you can show that that is also the
> same as the product of 1/(1 - p^-s)
> over all the primes p (primes this time
> not integers) which is a fascinating
> equivalence because the primes are
> very hard to deal with in formulae
> especially in a formula involving
> all the primes like this one.
>
> The way it works is that each 1/(1 - p^-s)
> there can be expanded into an infinite series
> and then when you take the product
> of all those infinite series in a
> careful way you find that they
> reconstitute all those
> terms in the original sum of the
> n^-s. I remember doing that as
> an undergraduate in my number theory
> classes but can't remember
> the details now.

It's quite simple and I have many books that explain it. Manfred
Schroeder's _Number Theory in Science and Communication_ is probably
the best. Let me know if you'd like be to post something on this --
though probably you can find it on the web.

> However, it says there that there is a connection
> with tuning. So I suppose that is
> the main interest here. Can anyone
> tell me, what is the connection
> of the zeta function with tuning
> anyone?

Gene's been discussing this for years. He must be responsible for
that blurb on Wikipedia. You can find the details on the tuning-math
list -- search for "zeta" . . .

🔗Robert Walker <robertwalker@...>

Hi Paul,

Just rmembered how it went and it was just a three
or four liner type proof:

To show that Sigma (1 to inf) 1/n^s
= Prod (1 to inf) 1/(1-p^-s)

1/(1-p^-s) = 1 + p^s + p^2s + ...

then multiplying those you are multiplying:
(1 + 2^s + 2^2s + 2^3s + ...)
(1 + 3^s + 3^2s + 3^3s + ...)
(1 + 5^s + 5^2s + 5^3s + ...)
...

and multplying all those together
you get all possible numbers of the
form
2^(k1*s) * 3^(k2*s) * 5^(k3*s) * ...
for all choices of k1, k2 and k3 positive non zero.
so by unique factorisation that gives all the
numbers and all of them once only
nice isn't it :-)

But you have to prove that both converge
in the same places which is the tricky part
if you want to do it properly - that
the product of all those series
converges at the same points
as the original sum.

Since the sum doesn't converge
for s = 1 (sum of 1/n the harmonic
series doesn't converge)
or for s real and less than 1
(e.g. n^-0.5 is larger than n^-1
for any n so the sum of the square
roots can't converge since the harmonic
series doesn't) then that's quite significant.

Here is a fun applet where you
can enter any complex starting
number, then do progressive
summation of the zeta function
and see where the result goes
http://www.math.ubc.ca/~pugh/Dirichlet/
:-).

Try it with different values along
the real = 0.5 line. Since it is real
valued along that line eventually it
should converge to some point along
the horizontal real line positive
or negative.

About Reimann:
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.html

Robert

🔗Gene Ward Smith <gwsmith@...>

--- In metatuning@yahoogroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> It's a little bit technical (you can easily look it up if you wish),
> but it is the #1 most important unsolved mathematical problem in
> existence. There is a huge edifice of mathematics whose truth is
> conditional on that of the Riemann hypothesis. A proof or disproof of
> the Riemann hypothesis would have a vast impact on the entire field
> of number theory as well as many related areas. Though a single
> counterexample hasn't come up in millions or billions of tests,
> that's no guarantee one won't -- a related hypothesis was believed
> true until a counterexample was found way up in the astronomical
> range (I think this is is Derbyshire's book).

It's not correct to say the Mertens Hypothesis, which is what I think
you are talking about, was generally believed to be true. Andy Odlyzko
thought like many others it was likely false, and used some heavy duty
computing to find a counterexample at a very, very large number.

Has this man proven the Riemann hypothesis?

🔗Carl Lumma <clumma@...>

🔗kraig grady <kraiggrady@...>

🔗Carl Lumma <clumma@...>

🔗Robert Walker <robertwalker@...>

🔗Paul Erlich <PERLICH@...>

🔗Paul Erlich <PERLICH@...>

🔗Robert Walker <robertwalker@...>

🔗Gene Ward Smith <gwsmith@...>

🔗Gene Ward Smith <gwsmith@...>

🔗Gene Ward Smith <gwsmith@...>

🔗Gene Ward Smith <gwsmith@...>