## Sunday, October 17, 2010

### Riemann Hypothesis Resolved Using Prime Numbers

The Riemann Hypothesis says that all the non-trivial zeros ( 0 ) are on the line ( y = ½ ) and that this hypothesis has something to do with prime numbers. Another way of expressing it, is by saying that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular, the error term in the prime number theorem is closely related to the position of the zeros. It is known that there are infinitely many zeros on the line 1/2 + it as t ranges over the real numbers. A prime number is any number that is evenly divided by itself and 1. Therefore, a prime number is a specialized real number which exists on the x axis. It is also known that any prime number can create a real number that isn’t a prime. A real number is a number that is sequentially placed on the x - axis and therefore has a specific location that can be calculated through subtraction. A prime number, however, cannot be specifically located on the x - axis because the distance between the prime numbers aren’t consistent. Riemann extended his equation into complex numbers which oscillate on a vertical plane at right angles to the x - axis or in other words across the line ( y = ½ ). The non-trivial zeros do not become real until the oscillating complex zero crosses ( y = ½ ) and becomes real. The complex zero is really on an imaginary string before it lies along the line ( y = ½ ). At this point the real zero can be used to calculate the location of the prime number on the x - axis. It is also known that there are an infinite number of prime numbers and by extension an infinite number of real numbers because prime numbers can be used to construct real numbers. So far the calculated zeros lie on the critical line, but the prime’s location strays because the calculation process behaves just like a 50:50 coin toss. The 50:50 distribution law says that coins will divide themselves proportionately over the long term, but on the specific terms they generally will not distribute themselves evenly. There are always exceptions. The most important thing that Riemann said was that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function and the error term in the prime number theorem is closely related to the position of the zeros.

Here are some facts about prime numbers.

1. Prime numbers, if they are prime numbers, have the numbers 1, 3, 7, 9 in column 0 ( farthest right column ).

2. If the sum of the digits of any number ending in 1, 3, 7, 9, total a multiple of 3, except for 3, ( for instance total 6, 9, 12, etc. ) it isn’t a prime number. If a number ending in 1, 3, 7, 9 in column zero (0), isn’t a prime number it can usually be evenly divided by a number with 1, 3, 7, 9 in column (0).

3. A prime number is defined as being only evenly divisible by itself and one ( 1 ).

Here’s the proof:

A prime number is defined as any number that can be only divided evenly by itself and one. Furthermore a prime number, if it is a prime number, only has the digits, 1, 3, 7, 9 in column 0 which is the farthest right column. I have also discovered that if the sum of the digits of any number ending in 1, 3, 7, 9 total a multiple of 3, except for prime number 3 ( for example total 6, 9, 12, etc. ) then it isn’t a prime number. The single digit prime numbers are 1, 2, 3, 5, 7, if we ignore the convention of no longer considering 1 as a prime number. A real number which includes prime numbers are located somewhere on the x - axis. If we multiply each single digit prime number ( 1, 2, 3, 5, 7 ) by ½ we get 1 at approximately ½, 2 at approximately 1, 3 at approximately 1.5, 5 at approximately 2.5 and 7 at approximately 3.5. 1 is actually at 1, 2 at 2, 3 at 3, 5 at 4 and 7 at 5. 97 is a prime and if we multiply 97 by ½, we get 48.5. The prime 97 isn’t anywhere close to being the 48th prime. The Riemann Hypothesis says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular, the error term in the prime number theorem is closely related to the position of the zeros. Our initial calculation indicated that the prime number 97 was oscillating around position 48.5 on the x axis which isn’t correct. The Riemann Hypothesis says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular, the error term in the prime number theorem is closely related to the position of the zeros. What to do???? We know that the prime number 97 lies on the x axis as it is a specialized real number. The Riemann Hypothesis says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. Therefore we make up a multiplier. The first digit is ½ or .5. The second digit is one of the Riemann Hypothesis zeros so we now have ( .50 ) . ( .50 ) X 97 is no better off than multiplying 97 X ( .5 ). We know that the prime numbers have 1, 3, 7, or 9 in column 0. Therefore arbitrarily add the 9 digit to ( .50 ) forming ( .509 ). If we multiply prime 97 X ( .509 ) the answer is worse. If, however, we take ( .509 ^ 2 ) we get ( .259081 ) and 97 X ( .259081 ) = 25.130857. Prime number 97 is in actuality the 26th prime. The Riemann Hypothesis says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular, the error term in the prime number theorem is closely related to the position of the zeros. Using the Riemann Hypothesis zeros between .5 and 9 ( .5----9 ) and raising it to a power we can oscillate the prime 97 around its’ position. Thus the error can be controlled by adjusting the Riemann Hypothesis zeros.

Here’s how the system works for numbers in general.

1. Count the number of digits in a prime number. For instance 7919 has 4 digits. Subtract 1 from the number of digits ( 4 - 1 = 3 ) for 7919. Form another number equal to the number of digits in 7919 ( 4 ) by putting ( .5 ) in the far left column and 9 in the far right column. ( .5—9 ). Fill the middle with Riemann Hypothesis zeros ( 0 ) forming a four digit number ( .5009 ). Raise ( .5009 ) to the power of 3 ( which is the number of digits in 7919 ( 4 ) minus 1 ( 4 - 1 = 3 ). ( .5009 ) ^ 3 = .125676215. Multiply 7919 X .125676215 which equals 995.2299524. 7919 is the 1000th prime. The answer is out by approximately 5. Adjust error accordingly using Riemann Hypothesis zeros.

As a matter of interest, if we are looking for the largest prime number in existence, the simplest way of doing it is to add digits to the left of any number ending in 1, 3, 7, 9 in column zero ( 0 ). Find the total of all the digits added together and divide by 3. If the result is an integer with no remainder, it is not a prime number. This truth can be verified by seeing if the number is only evenly divisible by itself and 1. It is also interesting that any number ending in 1, 3, 7, 9, if it isn’t a prime, is usually divisible by some number ending in 1, 3, 7, 9 in column 0.

If all the zeros used for calculating the position of primes on strings are real, then the primes themselves are real. Primes can be combined to create real numbers as well as fractions, so all the zeros for those numbers are real. Therefore the Riemann Hypothesis is true.

Actual Location Primes Multiplier 2 To Power 2 Power 2 Calculated Location
1 1 0.509999999 1 0.509999999 0.509999999
2 2 0.509999999 1 0.509999999 1.019999999
3 3 0.509999999 1 0.509999999 1.529999998
4 5 0.509999999 1 0.509999999 2.549999997
5 7 0.509999999 1 0.509999999 3.569999996
6 11 0.509999999 1 0.509999999 5.609999994
7 13 0.509999999 1 0.509999999 6.629999993
8 17 0.509999999 1 0.509999999 8.669999991
9 19 0.509999999 1 0.509999999 9.68999999
10 23 0.509999999 1 0.509999999 11.72999999
11 29 0.509999999 1 0.509999999 14.78999997
12 31 0.509999999 1 0.509999999 15.80999998
13 37 0.509999999 2 0.260099999 9.623699981
14 41 0.509999999 2 0.260099999 10.66409998
15 43 0.509999999 2 0.260099999 11.18429998
16 47 0.509999999 2 0.260099999 12.22469998
17 53 0.509999999 2 0.260099999 13.78529997
18 59 0.509999999 2 0.260099999 15.34589997
19 61 0.509999999 2 0.260099999 15.86609997
20 67 0.509999999 2 0.260099999 17.42669997
21 71 0.509999999 2 0.260099999 18.46709996
22 73 0.509999999 2 0.260099999 18.98729996
23 79 0.509999999 2 0.260099999 20.54789996
24 83 0.509999999 2 0.260099999 21.58829996
25 89 0.509999999 2 0.260099999 23.14889995
26 97 0.509999999 2 0.260099999 25.22969995

The position of the primes was calculated by multiplying the primes ( 1 to 31 ) by ( .509999999 ^ 1). From ( 37 to 97 ) the primes were multiplied by ( .509999999 ^ 2 ) to keep the calculated distance close to the actual distance.

Actual Location Primes String 1 String 2 Multiplier 2 To Power 2 Power 2 Calculated Location
27 101 2 2 0.509999999 2 0.2601 26.27009995
28 103 4 4 0.509999999 2 0.2601 26.79029995
29 107 8 8 0.509999999 2 0.2601 27.83069994
30 109 10 1 0.509999999 2 0.2601 28.35089994
31 113 5 5 0.509999999 2 0.2601 29.39129994
32 127 10 1 0.509999999 2 0.2601 33.03269993
33 131 5 5 0.509999999 2 0.2601 34.07309993
34 137 11 2 0.509999999 2 0.2601 35.63369993
35 139 13 4 0.509999999 2 0.2601 36.15389993
36 149 14 5 0.509999999 2 0.2601 38.75489992
37 151 7 7 0.509999999 2 0.2601 39.27509992
38 157 13 4 0.509999999 2 0.2601 40.83569992
39 163 10 1 0.509999999 2 0.2601 42.39629992
40 167 14 5 0.509999999 2 0.2601 43.43669991
41 173 11 2 0.509999999 2 0.2601 44.99729991
42 179 17 8 0.509999999 2 0.2601 46.55789991
43 181 10 1 0.509999999 2 0.2601 47.07809991
44 191 11 2 0.500999999 2 0.251001 47.9411909
45 193 13 4 0.500999999 2 0.251001 48.4431929
46 197 17 8 0.500999999 2 0.251001 49.4471969
47 199 19 1 0.500999999 2 0.251001 49.9491989

The position of the primes was calculated by multiplying the primes ( 101 to 181 ) by (.5099999999^2) From (191 to 199 ) the primes were multiplied by ( .5009999999 ^ 2 ) to keep the calculated distance close to the actual distance.

The most important thing that Riemann said was that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function and the error term in the prime number theorem is closely related to the position of the zeros. If you multiplied the primes from ( 191 to 199 ) by ( .509999999^2 ) the answer would be further from their real locations and hence the error would be greater.

It is also interesting that by either adding or subtracting Pi or ( e^1 ) depending on what is required you can get the Riemann Zero calculation closer to the actual location of the primes.

Actual Location Calculated Location Add / Subtract Total
1 0.509999999 0 0.509999999
2 1.019999999 0 1.019999999
3 1.529999998 0 1.529999998
4 2.549999997 0 2.549999997
5 3.569999996 0 3.569999996
6 5.609999994 0 5.609999994
7 6.629999993 0 6.629999993
8 8.669999991 0 8.669999991
9 9.68999999 0 9.68999999
10 11.72999999 0 11.72999999
11 14.78999997 -3.141592654 11.64840732
12 15.80999998 -3.141592654 12.66840733
13 9.623699981 3.141592654 12.76529263
14 10.66409998 3.141592654 13.80569263
15 11.18429998 3.141592654 14.32589263
16 12.22469998 3.141592654 15.36629263
17 13.78529997 3.141592654 16.92689263
18 15.34589997 3.141592654 18.48749262
19 15.86609997 3.141592654 19.00769262
20 17.42669997 3.141592654 20.56829262
21 18.46709996 3.141592654 21.60869262
22 18.98729996 3.141592654 22.12889262
23 20.54789996 3.141592654 23.68949261
24 21.58829996 3.141592654 24.72989261
25 23.14889995 0 23.14889995
26 25.22969995 0 25.22969995
27 26.27009995 0 26.27009995
28 26.79029995 0 26.79029995
29 27.83069994 0 27.83069994
30 28.35089994 0 28.35089994
31 29.39129994 0 29.39129994
32 33.03269993 0 33.03269993
33 34.07309993 0 34.07309993
34 35.63369993 0 35.63369993
35 36.15389993 0 36.15389993
36 38.75489992 -2.718281828 36.03661809
37 39.27509992 -2.718281828 36.55681809
38 40.83569992 -2.718281828 38.11741809
39 42.39629992 -3.141592654 39.25470726
40 43.43669991 -3.141592654 40.29510726
41 44.99729991 -3.141592654 41.85570726
42 46.55789991 -3.141592654 43.41630725
43 47.07809991 -3.141592654 43.93650725
44 47.9411909 -3.141592654 44.79959825
45 48.4431929 -3.141592654 45.30160025
46 49.4471969 -3.141592654 46.30560425
47 49.9491989 -2.718281828 47.23091707

It can be seen from these calculations that the magnitude of the oscillations of the primes around their expected position is controlled by the zeros ( 0’s) in the multiplier. The error term is closely related to the position of the zeros since ( .509999999 ) and (.599999990) both hold the same digits but the zero ( 0 ) position has been shifted thus putting the position of the primes further out of alignment. It can be seen that the calculation of the position of a prime and the number of primes preceding that prime all depend on the Riemann Hypothesis real zeros. Thus the Riemann Hypothesis has been proved by illustrating its’ real relationship to the position of the primes and the number of primes preceding it.