(1) ...... z(s)=
....................................s=x+jy........
x>1 ................... n=1
Some Extended Zeta Functions Provide Easy
Proofs of Riemann's Hypothesis
This page is now subject to the author's counterexample at http://www.coolissues.com/mathematics/Riemann/disproof.htm
James Constant
math@coolissues.com
While extended zeta functions support investigations of Riemann's hypothesis and estimates for the Prime Number Theorem, some zeta functions offer better prospects for providing easy proofs.
Definitions
A first zeta function is defined by
.................... oo
(1) ......
z(s)=
....................................s=x+jy........
x>1
................... n=1
A second zeta function is defined by
....................... oo
(2) ......z(1-s)=
...............................s=x+jy........
x<0
...................... n=1
In 1859, Riemann had the idea to define z(s) for all complex numbers s by analytic extension. This extension is important in number theory and plays a central role in the distribution of prime numbers.
There are a number of ways of extending the zeta
function to points where 0<x
1.1 One way of extending is by
using the
first f function alternating series defined by2
.................... oo
(3)
...... f(s)=
............................
s=x+jy ........
x>0
................... n=1
(4) ......
f(s)=(1-2
)z(s) .......................
x>1
A second f function is defined by
....................... oo
(5)
......
f(1-s)=
......................... s=x+jy ........
x<1
...................... n=1
(6) ......
f(1-s)=(1-2
)z(1-s) ....................
x<0
Equations (1) through (6) are analytic.
Riemann's Extended Zeta Function and Functional Equation
Euler (1737) noted that the formula
for integers
s>1 connected
integers and primes, the sum extending to all positive integers n,
and the product to all positive
primes
p. Considering 
as a function of z(s) of the
complex
variable s,
Riemann (1859) examined this
equation for s
as a complex number and
found that it can be extended to points with real part s
1 by the
formula (among others)3
which function is another form of an extended Riemann zeta function.
(9) ......
z(s) = 
z(1-s) ........
s=x+jy
which led him
to the theorem that
all the zeros of z(s), except those at s=-2,-4,-6,
. . . , lie in the critical
strip of the s-plane
for which 
where x
is the real part of s.
Riemann conjectured that all the zeroes in the critical strip
should lie on the critical line x=1/2.4 Attempts to prove or
disprove this
conjecture have generated a vast and intricate department of
analysis, especially since Hardy (1914) proved that z(s) has an
infinity of zeros on x=1/2.5 The question is still open
in 2001. A
prize is available to prove or disprove Riemann's hypothesis.6
the importance of which is that the f-functions can be easily expressed as complex numbers with real and imaginary parts. Elsewhere, I attempt a proof of Riemann's hypothesis using Riemann's extended zeta function (8).7, Here, I use extended zeta function (3) to make the proof.
It has already
been shown that
all zeros are in the critical strip 
and that
they are symmetric
about the critical line x=1/2.8 Riemann's functional
equation (9) can
be restated in terms of the new functional equation (10) in which
A(s)
0
at all points on the critical strip. Since functions f(s) and
f(1-s) are single valued at each point on the critical strip they
can be written in terms of their real and imaginary parts
f(s)=u+jv and f(1-s)=u'+jv' in which
.................. oo
..............................................
oo
(11)
......
u=
................v=-
................. n=1
............................................ n=1
.................. oo
.............................................
oo
..............u'=
................v'=
................. n=1
...........................................
n=1
When sin(ylnn)=0, equations (11) reduce to
.................. oo
(12)
...... u=
................v=0
................. n=1
.................. oo
..............u'=
..............
v'=0
................. n=1
(13)
yn,k=
, .....n=1,2,3,....,
.....k=0,1,2,3,....., sin(yn,klnn)=0,
......cos(yn,klnn)=(-1)k
.......................... oo
(14)
...... u=(-1)k
(-1)n+1
.........................
n=1
..........................
oo
............. u'=(-1)k
.......................... n=1
Equations
(14) are
conditionally convergent infinite series which, depending from
the rearrangement of terms, can be divergent or convergent to
some number including zero. In such series, the value of the sum
of the series can be changed at will by suitable rearraingement
of the series.9
Non-zero
values of each series occur in the
critical strip 0<x<1 even when x=1/2. Zero values mean that
the value of each
series, in the limit, approaches zero uà0 and u'à0.
Since the difference between the sum of a converging alternating
series and the sum of the first n terms is numerically less than
the (n+1)th term 
< 
and 
< 
. Thus, as nàoo
and uàu'à0,
xà1-x and xà1/2
which proves Riemann's hypothesis in the sense that, in the
limit, zeros of f(s)=f(1-s) approach but never reach the critical
line x=1/2.
A more rigorous proof puts zeros exactly on the critical line x=1/2 and nowhere else. Formally, since A(s) in functional equation (10) is determinate at all points of the critical strip including when x=1/2, f(s) and f(1-s) are different functions at all points of the critical strip except when x=1/2 and sin(ylnn)=0 for which f(s)=f(1-s)=0. This proves Riemann's hypothesis in the sense that zeros exist only on the critical line x=1/2. For example, if zeros exist on either side of and on x=1/2, then A(s)=f(s)/f(1-s)=0/0 and, thus, A(s) is indeterminate at all points of the critical strip including when x=1/2, a result precluded by determinate A(s) in equation (10). Note that, since sin(ylnn)=0, y is not a single point but a collection of double infinite numbers yn,k on the critical line. In summary:
thus proving Riemann's hypothesis.
The Prime Number Theorem and Riemann's Hypothesis
.................................................................................oo
(15) ......
p(x)=Li(x)+O(x2logx)
................... Li(x)=
................................................................................
2
where L(i) is Gauss's integral and O(x2logx) is an error term.10
Thus, if Riemann's hypothesis is true, equation (15) tells us that Li(x) is a very good approximation to p(x). The known values of p(x) today agree with equation (15) but it is admitted that it is not easy to "have an idea of what should be exactly the error term." 11. Here, results (13) and (14) should be helpful.
1 See E.C. Titchmarsh, The theory of the Riemann Zeta-function, Oxford Science Publications, second edition, revised by D.R.Heath-Brown (1986).
2 See R. Courant, Differential and Integral Calculus, Interscience Publishers, Inc. New York 1947 Vol. II page 568 problem 13.
3 See "The Rieman Hypothesis in a Nutshell" at http://www.math.ubc.ca/~pugh/RiemannZeta/RiemannZetaLong.html
4 See Chris Caldwell's "The Riemann Hypothesis" (University of Tennessee at Martin) at http://www.utm.edu/research/primes/notes/rh.html
5 See E.T. Bell, The Development of Mathematics, Dover Publications, New York 1972. Page 315.
6 See Enrice Bombieri's "The Riemann Hypothesis" (Clay Mathematics Institute) at http://www.claymath.org/prize_problems/riemann.htm
7 See my Proof of Riemann's Hypothesis at http://riemann.coolissues.com/rieman.htm
8 See endnote 4 above.
9 See R. Courant, Differential and Integral Calculus, Interscience Publishers, Inc. New York, 1947 2d ed. Vol. I pages 372-375.
10 See pages 5-7 in Chris Caldwell's "How Many Primes Are There" at http://www.utm.edu/research/primes/howmany/shtml
11 See "The Riemann's Zeta Function z(s)" at http://numbers.computation.free.fr/Constant/Primes/counting/Primes.html
Copyright 
by James Constant 2003
By the same author: http://www.coolissues.com/mathematics/sameauthor.htm
1 by means of the formula 
1 by means
of the formula 
f(1-s),
except when x=1/2,
functional equation (10) precludes zeros in the critical strip,
except when x=1/2. I now show that f(s)=f(1-s)=0 only on the
critical line x=1/2.