Riemann Hypothesis according to Bearnol

bearnol · 2006-07-29, 10:50

http://www.bearnol.pwp.blueyonder.co...th/riemann.htm

Fusion_power · 2006-07-29, 12:54

Methinks bearnol's "solution" suffers from chicken or egg syndrome.

The last serious attempt at a Reimann proof was from de Branges about a year ago. There are several attempts recently published but most are deficient in one or another aspect. A google search for "Riemann Branges" will bring up some interesting reading.

Fusion

bearnol · 2006-07-29, 14:01

My solution is correct and complete - think about it for more than 2 seconds and you'll see. There is some surprisingly clever logic hidden in those few lines :)
btw, it _also_ demonstrates pretty clearly - in rebuff to your assertion of chicken and egg - why the zeros of zeta occur in symmetric pairs about the real axis, a fact I was not even aware of until after I came up with the proof, but which I later (also) read to be true...
J

jinydu · 2006-07-29, 18:17

Also, bearnol's "solution" contains lots of indecipherable notation...

bearnol · 2006-07-29, 19:43

It uses, I believe, one of a number of fairly standard ways of expressing some mathematical notation (eg superscripts) in plain ASCII. I deliberately chose this format (ie plain text) (a number of years ago) to try and make the proof as accessible as possible to as wide a range of browsers as possible.
Since the proof's exposition (if not inspiration) is so simple, anyone knowing the form of the zeta function should be able to follow it very easily, regardless of exact coding anyway.
J

akruppa · 2006-07-29, 21:21

Is that supposed to be

$\left( \sum_{n=1}^{\infty} n^{-s} \right) \cdot n^{2s-1}$

or

$\sum_{n=1}^{\infty} \left( n^{-s} \cdot n^{2s-1} \right)$

Looks like you're multiplying the expression by n^{2s-1}, even though n is the variable of summation. Then you pull that multipler into the sum even though it's not constant.

Is that what you were trying to do?

Alex

jinydu · 2006-07-29, 22:11

Quote:

Originally Posted by jinydu

Also, bearnol's "solution" contains lots of indecipherable notation...

For example, "n**-s"

Besides, $\zeta(s)\neq\zeta(1-s)$ in general. For example, $\zeta(1)\neq\zeta(0)$ .

bearnol · 2006-07-30, 07:43

Quote:

Originally Posted by akruppa

Is that supposed to be

$\left( \sum_{n=1}^{\infty} n^{-s} \right) \cdot n^{2s-1}$

or

$\sum_{n=1}^{\infty} \left( n^{-s} \cdot n^{2s-1} \right)$

Alex

The latter

bearnol · 2006-07-30, 07:48

Quote:

Originally Posted by jinydu

Besides, $\zeta(s)\neq\zeta(1-s)$ in general. For example, $\zeta(1)\neq\zeta(0)$ .

True, zeta(s) <> zeta(1-s) in general. My proof only applies when zeta(s) = 0. Then zeta(s) DOES = zeta(1-s)

akruppa · 2006-07-30, 09:22

> The latter

Where does the n^(2s-1) term come from?

> My proof only applies when zeta(s) = 0. Then zeta(s) DOES = zeta(1-s)

The trivial zeros are not symmetric around s=1/2.

Alex

EDIT2: All posts but first moved to separate thread.

bearnol · 2006-07-30, 10:29

Quote:

Originally Posted by akruppa

> The latter

Where does the n^(2s-1) term come from?

I _choose_ the 2s-1 term. It is chosen for two reasons:
1) It 'converts' the s term into 1-s
2) because the real part of s is less than 1, the real part of 2s-1 is also less than 1. Hence the summation must still sum to zero

Quote:

Originally Posted by akruppa

EDIT2: All posts but first moved to separate thread.

Thanks

2006-07-29, 10:50	#1
bearnol Sep 2005 127 Posts	Proved several years ago! http://www.bearnol.pwp.blueyonder.co...th/riemann.htm

2006-07-30, 09:22	#10
akruppa "Nancy" Aug 2002 Alexandria 2,467 Posts	> The latter Where does the n^(2s-1) term come from? > My proof only applies when zeta(s) = 0. Then zeta(s) DOES = zeta(1-s) The trivial zeros are not symmetric around s=1/2. Alex EDIT2: All posts but first moved to separate thread. Last fiddled with by akruppa on 2006-07-30 at 10:09 Reason: 1/2, not 1

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2006-07-29, 12:54	#2
Fusion_power Aug 2003 Snicker, AL 7×137 Posts	Methinks bearnol's "solution" suffers from chicken or egg syndrome. The last serious attempt at a Reimann proof was from de Branges about a year ago. There are several attempts recently published but most are deficient in one or another aspect. A google search for "Riemann Branges" will bring up some interesting reading. Fusion

2006-07-29, 14:01	#3
bearnol Sep 2005 127 Posts	My solution is correct and complete - think about it for more than 2 seconds and you'll see. There is some surprisingly clever logic hidden in those few lines :) btw, it _also_ demonstrates pretty clearly - in rebuff to your assertion of chicken and egg - why the zeros of zeta occur in symmetric pairs about the real axis, a fact I was not even aware of until after I came up with the proof, but which I later (also) read to be true... J

2006-07-29, 18:17	#4
jinydu Dec 2003 Hopefully Near M48 1758₁₀ Posts	Also, bearnol's "solution" contains lots of indecipherable notation...

2006-07-29, 19:43	#5
bearnol Sep 2005 127 Posts	It uses, I believe, one of a number of fairly standard ways of expressing some mathematical notation (eg superscripts) in plain ASCII. I deliberately chose this format (ie plain text) (a number of years ago) to try and make the proof as accessible as possible to as wide a range of browsers as possible. Since the proof's exposition (if not inspiration) is so simple, anyone knowing the form of the zeta function should be able to follow it very easily, regardless of exact coding anyway. J

2006-07-29, 21:21	#6
akruppa "Nancy" Aug 2002 Alexandria 2,467 Posts	Is that supposed to be $\left( \sum_{n=1}^{\infty} n^{-s} \right) \cdot n^{2s-1}$ or $\sum_{n=1}^{\infty} \left( n^{-s} \cdot n^{2s-1} \right)$ Looks like you're multiplying the expression by n^{2s-1}, even though n is the variable of summation. Then you pull that multipler into the sum even though it's not constant. Is that what you were trying to do? Alex