PROOF OF RIEMANN'S HYPOTHESIS

James Constant

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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

. . . . . . . . . . . . . . . . . . . . . . .x>1 . . . . . . . . . . . . . . . . . (1)

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 the complex variable s, Riemann (1859) proved that satisfies a functional equation

. . . . . . . . . . . . . . . .. . . . (2)

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 2008. A prize is available to prove or disprove Riemann's hypothesis.3

Finding Zeros Using Riemann's Zeta Function

When extended to values in the critical strip Riemann's zeta function is written as

. . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . (3)

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 I will now show that all zeros are on the critical line x= ½ and that functional equation (2) presents a problem.

Riemann's functional equation can be restated as in which  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 partsand in which

. . . . . .

. . . . . . . . . . . k=lnn            (4)

in which k=lnn is the natural logarithm of n. Note that k is an irrational number.

On the critical line x= ½ and 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 and . Thus, if =0 in the critical strip where then, since and , 0 and Riemann's functional equation cannot be satisfied. Riemann's functional equation, therefore, precludes zeroes at points where in the critical strip. All zeroes in the critical strip are on the critical line x= ½.

When , m=0,1,2, . . . equations (4) reduce to

. . . . . . . . . . k=lnn . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .(5)

in which u=u'=0 when x=1/2, since infinite series are conditionally converging series which can be made to converge to zero by a suitable rearrangement of terms. Accordingly, on the critical line x=1/2 when , m=0,1,2, . . . Note that since k is an irrational number y is a rational number. See ADDENDUM for another way of finding zeroes of .

Zeroes of the Riemann zeta function-The Functional Equation Problem

The Riemann zeta function has zeroes at the negative even integers. These are called the trivial zeroes. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(os/2)being 0 in the functional equation. The non-trivial zeroes have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip , which is called the critical strip. The Riemann hypothesis, considered to be one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has x = 1/2. In the theory of the Riemann zeta function, the set when x = 1/2 is called the critical line.

The location of the Riemann zeta function's zeroes is of great importance in the theory of numbers. From the fact that all non-trivial zeroes lie in the critical strip one can deduce the prime number theorem. It is known that there are infinitely many zeroes on the critical line. And, directly from the functional equation (2), one sees that the non-trivial zeroes are symmetric about the axis x=1/2. Furthermore, the fact that for all complex (~ indicating complex conjugation) implies (emphasis intended) that the zeroes of the Riemann zeta function are symmetric about the real axis.

Such reliance on functional equation (2) is not warranted. Essentially, functional equation (2) says that values of the zeta function at s can be computed from its values at 1-s, i.e., for each non trivial zero at 1-s, the value of s is also a zero of . I find zeroes by using equations (4) and (5). Not withstanding functional equation (2), is a necessary but not sufficient condition for finding the value of . The reason is simple. As expressed in (4) and (5), (3) is a conditionally converging series which can be made to converge to any number value by a suitable selection of terms. Knowing , therefore, does not necessarily establish .

Unless x=1/2, series u and u' are different series. Depending from the way terms are selected. each series has many possible values, including zero. It follows that at each point x,y in the critical strip, and on the critical line, the value of is unknown. There is no symmetry about the critical line. However, there is symmetry when x=1/2. We can say for sure that on the critical line the value, including zero, of which appears at +y is the same value at -y.

The sufficiency, therefore, of functional equation (2) is obtained when u=u' in equation (5), i.e., when x=1/2. All zeroes are located on the critical line x=1/2 when

. . . . . m=0,1,2, . . .         k=lnn    . . . . . . . .                          (6)

from which I conclude ny is a rational number. This can occur only if y is an integer or, if y=p/q is a rational non integer when n=kq, where p,q,k are integers. In either case, y is a rational number. Thus,

. . . . . . . . . .             (7)

which says zeroes of exist on the critical line at rational number locations y=p/q when n=kq.

Connecting Critical Line Zeroes and Prime Numbers

The Prime Number Theorem (PNT) states that the yth prime py is of the order (~) of ylogy or that the number of primes ~ y/logy. A consequence of the PNT is that which says that we can find knowing py, or find py knowing , within some order of magnitude. The PNT was proved by Hadamard and de la Vallee Pousson (independently) using Riemann's Hypothesis, after showing that the zeroes of Riemann's zeta function cannot lie too far off the critical line. It is now known that Riemann's Hypothesis produces the result =Li(y)+O(ylny) where Li(y)=is Gauss's integral and the O term is the order of the error.5 It is well known, therefore, that the PNT is an approximate predictor of the number of primes in any interval y

In the present proof, Riemann's zeta function on the critical line when      k=lnn,   m=0,1,2, . . . Since y is a single dimensional number, n=n(m) and

m=0,1,2, . . .                                   . . . . . . . . . . .  . . . .      (8)

which gives the number of zeroes  of   in the interval m located on the critical line. If each side of equation (8) is multiplied by

m=0,1,2, . . .                                   . . . . . . . . . . .  . . . .      (9)

then  in which , in view of equation (9), is a prime number.

Equations (8) and (9), therefore, give
the number of prime numbers or zeroes  of    in the interval m located on the critical line and the mth prime number or zeroe, respectively.  Interestingly, equation (9) shows that the ratio of irrational numbers    is a rational number   in which  is a prime number. Note that in the PNT notation =.

Since function n(m) is not known, it will be necessary to provide estimates for equations (8) and (9). Fraction 1/lmn(m) in equation (8) is known  as a descending function of m. Consider replacing it in equations (8) and (9) by a best fit nonlinear regression curve

a=0.062095933    b=0.86509468      (10)

in which, by way of example, values of a and b were obtained using 9 values of m and
available in the literature. See http://www.coolissues.com/mathematics/Fprimes/fprimes.htm

ADDENDUM - Finding Zeroes Using Taylor Series

There are several ways of finding zeroes of . In the foregoing, I use infinite series (3) which can be made zero in two ways, first, by finding the limits of the entire series and, second, by finding each term is zero. In doing the latter, I find equations (5) represent and when , m=0,1,2, . . .

Since is an analytic function at a point so, another way of finding its zeroes is by expanding it into a Taylor's series and finding that all its derivatives are zero

. . . . . . . . . . . .. . . . . .                (11)

in which is the n'th derivative of . Again, (11) can be made zero in one of two ways. Here, I find that all derivatives are zero in the same manner was previously equal to zero. When so is a zero of , and the sum in (11) are zero. Accordingly, since

. . . . . . . . . . . . . . . m=0,1,2, . . . ,     n'=1,2, . . . . . . . . . .(12)

and, .

1 Chris Caldwell The Riemann Hypothesis (University of Tennessee atMartin) 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 Enrice Bombieri's The Riemann Hypothesis (Clay Mathematics Institute) at h ttp://www.claymath.org/prize_problems/riemann.htm

4 Caldwell note 1 above

5 Chris Caldwell "How Many Primes Are There" pages 5-7 at http://www.utm.edu/research/primes/howmany/shtml