PROOF OF RIEMANN'S HYPOTHESIS
James Constant
math@coolissues.com
Riemann's hypothesis is proved using Riemann's functional equation.
Introduction
The famous conjecture known as Riemann' s hypothesis1 is to classical analysis what Fermat's last theorem is to arithmetic. Euler (1737) noted that the formula
the sum extending to all positive
integers n, and the product to all positive
primes p. The necessary conditions of
convergence hold for complex values of s
with real part >1. Considering 
as a function of 
of the complex
variable s, Riemann (1859) proved that 
satisfies a functional
equation
which led Riemann to the theorem that
all the zeros of 
, except those at s=-2,-4,-6, . .
. , lie in the strip of the s-plane
for which 
where
x is the real part of s.
Riemann conjectured that all the zeros in the strip should lie on
the line x= ½. Attempts to prove or
disprove this conjecture have generated a vast and intricate
department of analysis, especially since Hardy (1914) proved that
has an
infinity of zeros on x= ½ .2
The question is still open in 2000. A prize is available to prove
or disprove Riemann's hypothesis.3
Proof Using Riemann's Functional Equation
It has already been shown that all
zeros are in the critical strip 
and that they are
symmetric about the critical line x= ½.4
Riemann's functional equation can be restated as 
=A(
in which A(
0 at all
points in the critical strip. Since functions 
and 
are single valued at
each point in the critical strip they can be written in terms of
their real and imaginary parts 
=u + iv and
=u'
+ iv' in which
On the critical line x= ½
and 
=A(
in which s~
is the conjugate of s. Thus, if 
=0 on the critical line
then , since u=u'=0 and v=v'=0, 
=0
and Riemann's functional equation is satisfied. At all other
points in the critical strip x 
½ and
A(
. Thus, if 
=0 in the critical
strip where x 
½ then, since u
u'
and v
v', 
0 and
Riemann's functional equation cannot be satisfied. Riemann's
functional equation, therefore, precludes zeros at points where
x 
½
in the critical strip. All zeros in the critical strip 
are on the
critical line x 
½.
1 See Chris Caldwell's The Riemann Hypothesis (University of Tennessee at Martin) at http://www.utm.edu/research/primes/notes/rh.html
2 E.T. Bell, The Development of Mathematics, Dover Publications, New York 1972. page 315.
3 See Enrice Bombieri's The Riemann Hypothesis (Clay Mathematics Institute) at http://www.claymath.org/prize_problems/riemann.htm
4 See note 1 above.
Copyright © 2003 by James Constant
By the same author:
Proof of Fermat's Last theorem at http://fermat.coolissues.com/fermat.htm
Is Fermat's Last Theorem Proven? at http://wiles.coolissues.com/wiles.htm
Some Extended Zeta Functions Provide Easy Proofs of Riemann's Hypothesis at http://zeta.coolissues.com/zeta.htm
Finding Prime Numbers at http://fprimes.coolissues.com/fprimes.htm
Algebraic Factoring of the Cryptography Modulus and Proof of Goldbach's Conjecture http://goldbach.coolissues.com/goldbach.htm
Proof of the Twin Primes Conjecture http://tprimes.coolissues.com/tprimes.htm
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