Conference rusure::math

1289.0. "Louis de Branges may have proven the Riemann Hypothesis" by NOEDGE::HERMAN (Franklin B. Herman DTN 291-0170 PDM1-1/J9) Thu Aug 23 1990 19:50

	I was just forwarded this off the usenet. If this is reliable
    and Louis de Branges has indeed solved the Riemann Hypothesis, then
    this would be the biggest mathematical event of the century, 
    the equivalent in physics in terms of depth and importance to 
    Einsteins Theory of Special and General Relativity. 

	For those not familiar with the Riemann Hypothesis, here is
    a complete statement:

	Define
                        1         1              1   
        Zeta(s) = 1 + -----  +  ----- + ... +  ----- + ...
                        s         s              s
                       2         3              n    

                       1             1                  1         
                =  ---------- *  ---------- * ... *  ---------- * ... 
                   (1 - 2^-s)    (1 - 3^-s)          (1 - p^-s)     

    where the infinite product runs over all primes p and the sum and
    product converge absolutely for real s > 1. Letting s be a 
    complex number, Riemann observed that both still converged for
    s with real part > 1 and from the product expansion (due to Euler),
    has no zeros in the region of convergence Re (s) > 1. 

	Riemann proved that via a functional equation, the definition
    of Zeta(s) could be extended recursively to the entire complex plane 
    with a unique pole of order one at s=1. Specifically, define the
    Gamma function which is the complex analysis generalization of the
    factorial function by the following improper integral:
                         __ 
                        / t=oo   t  s-1
	Gamma(s) =     /        e  t   dt
                    __/ t=0

    This converges for complex s with Re(s) > 0. Also

	Gamma(n+1) = n! for n=0,1,2,...

    and the Gamma function satisfies the functional equation:

	Gamma(z+1) = z Gamma(z)

    This allows one to extend the definition of Gamma to a meromorphic 
    function on the entire complex plane with simple poles at the 
    negative integers.

	Next define

		 -s/2
	Z(s) = PI    Gamma(s/2) Zeta(s)

    Then finally, Zeta satisfies the functional equation:

	    Z(s) = Z(1-s)

    Using this functional equation, its not that hard to show that the 
    only zeros of Zeta for the left half space Re(s) <= 0 are at 

	    s = -2, -4, -6, ...

    However, Zeta does have an infinite number of zero's in the strip
    0 < Re(s) < 1 called the "non-trivial zeros" of Zeta. B. Riemann's
    Hypothesis is that ALL the non-trivial zeros of Zeta lie on the
    the line Re(s) = 1/2. 
    
	-Franklin

--------------------------------------------------------------------------------
From usenet.ins.cwru.edu!tut.cis.ohio-state.edu!ucsd!usc!zaphod.mps.ohio-state.
edu!shape.mps.ohio-state.edu!edgar Thu Aug 23 16:39:11 EDT 1990
Article 7625 of sci.math:
Path: usenet.ins.cwru.edu!tut.cis.ohio-state.edu!ucsd!usc!zaphod.mps.ohio-state.
edu!shape.mps.ohio-state.edu!edgar
>From: [email protected] (Gerald Edgar)
Newsgroups: sci.math
Subject: De Branges again
Message-ID: <[email protected]>
Date: 23 Aug 90 17:34:52 GMT
Sender: [email protected]
Organization: The Ohio State University, Dept. of Math.
Lines: 10
 
 
Remember Louis de Branges, solver of the Bieberbach conjecture.
He has announced a talk
  "A Proof of the Riemann Hypothesis"
at the Wabash Seminar, September 8.  Who knows something about this?
--
  Gerald A. Edgar          
  Department of Mathematics             Bitnet:    EDGAR@OHSTPY
  The Ohio State University             Internet:  [email protected]
  Columbus, OH 43210   ...!{att,pyramid}!osu-cis!shape.mps.ohio-state.edu!edgar
 
 

	It would be great to have this finally settled.  I believe
	there are also a number of interesting results of the form
	"if Riemann's Hypothesis is true, then ...".  Thanks for
	the full statement of RH.  Is there also a "Generalized
	Riemann's Hypothesis" and if so do you know what it is?

>>	Remember Louis de Branges, solver of the Bieberbach conjecture.

	What is the statement of the Bieberbach conjecture that he solved?

	Dan

    Re: -1:

>>  What is the statement of the Bieberbach conjecture that he solved?

	Here's a complete statement with a little motivation:

	A univalent function is a 1-1 holomorphic function defined on the
    open disk D = {z: |z| < 1}. Via a composition with linear fractional 
    isometry of D, one can normalize any univalent function, f(z) say, by
    
	    f(0) = 0 and f'(0) = 1
    
    With this normalization its not difficult to prove the following
    "distortion inequalities":

           |z|                          |z|        
        ---------    <= |f(z)|  <=   ---------     
                 2                            2    
        (1 + |z|)                    (1 - |z|)     

 
		    and 

           |z|                           |z|        
        ---------    <= |f'(z)|  <=   ---------     
                 3                             3    
        (1 + |z|)                     (1 - |z|)     


    with equality holding only if 
                                    n=oo           
                       |z|          ____    n-1 n  
            f(z) =  ---------    =  >,   n e   z   , where |e| = 1
                             2      ----           
                    (1 - ez)        n=1            



    From this Bieberbach conjectured [1916] that given a normalized univalent
    function f(z) with Taylor series expansion:

                    n=oo           
                    ____        n     
        f(z) = z +  >,    a[n] z   
                    ----           
                    n=2           

    
    then

	    |a[n]| <= n , n = 2,3,...


    -Franklin

    Anyone heard anything new on this?
    
    Eugene

    Re .0:
    
>	I was just forwarded this off the usenet. If this is reliable
>    and Louis de Branges has indeed solved the Riemann Hypothesis, then
>    this would be the biggest mathematical event of the century, 
    
    While I will grant you that it would be an incredibly important event,
    it would fall somewhat short of the mathematical event of *this*
    century.
    
    That title will, once the century is over, almost certainly go to
    the resolution of another question on Hilbert's program.  (Though of
    course no one could prove that ;-)
    
    --  Barry