HEIGHT=60 WIDTH=275> 
       

The Riemann zeta function has the trivial zeros at -2, -4, -6,
        ... (the poles of 
SRC="http://primes.utm.edu/gifs/gamma.gif" ALT="gamma" >(s/2)).  Using the Euler product
        (with the functional equation) it is easy to show that all the other
        zeros are in the critical strip of non-real complex numbers with
        0 <        Re(s) < 1, and that they are symmetric
        about the critical
        line Re(s)=1/2.  The unproved Riemann hypothesis is
        that all of the nontrivial zeros are actually on the critical line. 
     

In 1986 it was shown that the first 1,500,000,001 nontrivial zeros of
        the Riemann zeta function do indeed have real part one-half [VTW86].  Hardy
        proved in 1915 that an infinite number of the zeros do occur on the critical
        line and in 1989 Conrey showed that over 40% of the zeros in the critical
        strip are on the critical line [Conrey89].  However, there is
        still a chance that the Riemann hypothesis is false.  From August
        of 2001 through 2005, Sebastian Wedeniwski ran ZetaGrid which
      verified that the first 100 billion zeros were on the critical line.      
     

2. Who cares?      

In 1900 Hilbert listed proving or disproving this hypothesis as one
        of the most important unsolved problems confronting modern mathematics
        and it is central to understanding the overall distribution of the primes. 
When Hadamard and de la Vallee Poussin proved the prime
          number theorem, they actually showed

for some positive constant a, and they did this by bounding the
        real part of the zeros in the critical strip away from 0 and 1.  The
        error term is directly dependent on what was known about the zero-free
        region within the critical strip.  As our
        knowledge of the size of this region increases, the error term decreases.         In
        fact, in 1901 von Koch showed that the Riemann hypothesis is equivalent
        to

            WIDTH=197>       

 There are many results like this, see, for example [[url=../references/refs.cgi/BS96]BS96[/url]].      
     

Generalizations of RH

Recall again our starting point from Euler:

          ALT="zeta(s) = 1 + 2^(-s) + 3^(-s) + 4^(-s)..." HEIGHT=60 WIDTH=300>

Why should the numerators all be one?   One important
        way to alter the series is to replace the numerators with functions χ(n)
        called Dirichlet characters (these can be viewed as functions for which
        there exists a positive integer k with χ(n + k)
        = χ(n)
        for all n, and with χ(n) = 0 whenever gcd(n, k) > 1).   The
        resulting infinite sum L(?,s) is a Dirichlet L-function.   Once
        again we analytically continue the function to one that is meromophic
        on the entire complex plane.   The extended
      Riemann Hypothesis is that for every Dirichlet character χ and
      the zeros L(χ,s) = 0 with 0 < Re(s) < 1, have
      real part 1/2.  The distributions of the zeros of these L-functions
      are closely related to the number of primes in arithmetic progressions
      with a fixed difference k.   Should the extended Riemann Hypothesis
      be proven, then [url=../prove/prove2_3.html]Miller's
      test[/url] would provide an efficient primality proof for general numbers.       See,
      for example, [[url=../references/refs.cgi/BS96]BS96[/url] 8.5-6].

Another way to generalize Euler's sum is to leave the
    field of rational numbers, and replace the denominators with the norms of
    the non-zero ideals (special sets of elements) in a finite field extention
    of the rationals K (called
    a number field).   The resulting sum is the Dedekind zeta-function of K and
    can again be analytically continued.   These zeta functions also have
    a simple pole at zero and infinitely many zero in the critical region.  The generalized
    Riemann Hypothesis    is again that the zeros in the critical region
    all have real part 1/2.  See, for example, [[url=../references/refs.cgi/BS96]BS96[/url] 8.7].