The Riemann Hypothesis

ABSTRACT It has been shown in the solution of the Riemann Hypothesis (RH) presented in this document that the RH and the Prime Number Theorem (PNT) are solidly linked. The PNT was proved in 1896 by Jacques-Solomon Hadamard and Charles Jean de la ValleePoussin, and is universally accepted by mathematicians. The relationship between the RH and the PNT values, as illustrated in this document in Tables (2&3), is so close, the RH values might have been more effective in the proof of Chebyshev’s Conjecture for the PNT instead of vice versa. In both the RH and the PNT, the zeros (primes) of the RH and the primes of the PNT produce the same result within a value of 1.6 % for the entire counting number line. They both clearly show that the zeros (primes) of the RH are infinite and lie on the “critical line”, and the primes of the PNT are infinite and also lie on the same line. The conclusion that the RH is true, perforce, must be accepted; or alternatively, the PNT must be rejected as false. In addition, the role of the principle of “The Excess of Nines” on both the RH and PNT is illustrated. I have read numerous articles concerning the RH and the PNT and have yet to find any mention of this principle, which I believe is critical to the solutions of both. It is this principle that gives the beginning, mid, and terminal values of each cycle as defined in this document; precluding the existence of Ref (10) zeros at these specific points since the excess of nines = 1 at each. (see Section VII). A simple ratio which I chose to call Martin’s Ratio permits converting PNT values to RH values or vice versa and then to replicate any of the Ref (10) zeros with either; the only variant being cycle number. Martin’s Ratio provides a clear theorem approach to the proof of the RH by utilizing the PNT as a corollary proof. Mack Lendon Martin (July 10, 2007)

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THE RIEMANN HYPOTHESIS By Mack L. Martin I Historical Background In 1859 Bernhard Riemann introduced complex numbers into the zeta function of Leonhard Euler and the function took on the name of the Riemann zeta function. Riemann conjectured that the prime numbers were intimately linked with his conjecture, 1 and all the zeros of the function would lie on the “critical line” at x = . The function, 2 first introduced: ∞

ζ ( s ) = ∑ n− s n =1

(1.1) by Euler in 1740, where s was confined to the interval s>1 for all real numbers. Euler also showed the equivalence of the zeta function to the Euler Product Formula, which he developed: −1

 1  ζ ( s ) = ∏ 1 − s  (1.2) pn  n =1  Riemann’s conjecture became known as the Riemann Hypothesis (RH), but needed to be analytically continued to be valid in the range 0<x<1 for the complex number s = x + it 1 or more specifically, for, s = + it . Since the notation for complex numbers used by 2 engineers and scientists is z = x + iy , from hereon the complex number used to determine 1 the zeros will be noted as z = + it which will not cause any difficulty. The analytic 2 continuation of the Riemann zeta function will then be: ∞

ζ ( z) =

1 1 − 21− z

∞

∑ n =1

( −1) nz

n −1

(1.3)

This equation extends the validity of the Riemann zeta function to the interval of interest: 0 < Re < 1 The zeta function in this form will produce zeros for all values found on the website, Ref (10). Much of the above information can be found in numerous sources on the web, but most of that which follows was a discovery of the author, since during all of the research I have done, I have not seen a single reference to the techniques I have used to solve the RH.

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II Analysis The initial approach that I have taken in the solution of the RH is geometric/mathematical that provides a better insight into its structure. This approach led to the theorem statement in Section VIII. The operative function that I have investigated is the form shown in equation (1.3), repeated below for convenience, which I decided to analyze as two separate terms. This causes no concern since the first term merely multiplies the summand and its absolute value is always within the interval 2 − 1 ≤ M ≤ 2 + 1 : ∞ 1 ( −1) ζ ( z) = ∑ 1− z 1 − 2 n=1 n z 1 ….is defined as the M-Component. 1 − 21− z n −1 ∞ ( −1) ….is defined as the Z-Component. ∑ nz n =1 n −1

(1.3) repeated.

The M-Component is vital to the solution I will present, beyond its primary purpose of making the zeta function analytic in the interval 0<x<1, it has ramifications that extend to the Prime Number Theorem (PNT) proven by Hadamard and Vallee-Poussin in 1896. While experimenting with the M-Component, using Ref.(9), I discovered that for odd 1 values of k inputted into z = + it , where t was computed with the algorithm 2 k t = ⋅ ( 9.064720283654 ) , the M-Component evaluated to 0.414; when even numbers 2 were used it evaluated to –(2.414). This was true for any odd or even number used on the natural number line and I had no idea why this strange expression caused the MComponent to behave in this manner. Then once, while performing an operation on the 2π zeta function with Ref (9), I noticed that an exponent of 2 was returned as . It was ln ( 2 ) apparent that this ratio might evaluate to the above large number; and it did exactly. This was a serendipitous discovery that I wish I could say came about constructively. At any rate, this discovery set the course for nearly all of my subsequent activity. Since the assumption has already been made that the real part of the complex number z to 1 be used is x = , the M-Component can be put into a form that is dependent only on the 2 imaginary part t substituted into the M-Component for z in:

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1

M= 1−

(1.4)

2 it

2 After first tailoring, without changing its value, to have our newfound term (which for lack of a better descriptor, I will call the period cycle) accept values of t in terms of k. k  2π t =  2  ln ( 2 )

  

k = 1, 2,3.....n of the natural number line.

k 0 1 2 3

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M

( ( ( ( (

) 2 − 1) 2 + 1) 2 − 1) 2 + 1) 2 +1

t=

(1.5)

k 2π 2 ln ( 2 ) 0

4.53236014 9.06472028 13.59708043 18.12944057

The first few values of t, as well as the absolute values of M, are shown in the table above for the beginning values of k. As indicated, the maximum absolute value of M is 2.414 and the minimum absolute value is 0.414. It will be proven that a zero of the zeta function is constrained to occur only at a fractional value of k, and this will have an affect extending into the Prime Number Theorem. It is advantageous to plot equation (1.4) in order to analyze its geometric construction. The software uses a range variable to generate the points for the plot, and if it is too large, it is almost impossible to follow the manner in which the plot is constructed. The range variable determines the number of points t to plot; however, to see the odd and even behavior of the plot, it is necessary to plot the t values prior to multiplication by the constant of the period cycle. Using a very large range variable, the red-colored figure in the Fig (1) will take on a circular geometry with a radius of 1.414 and centered at (-1, 0). The mirror image (blue) was included to enhance the symmetry of the figure, but it is unnecessary to explain it. In order to reveal exactly how the figure is generated, the range variable must be considerably reduced and the mirror image figure removed. When this is done, and Fig (2) is referenced, the letter groupings start to the left of the graphic beginning with A at 0, and continue counter-clockwise around the graphic with each

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column of numbers, in sequence with the natural numbers, until grouping I is reached at 8, or until 1 cycle is completed at k = 9 again at A. The procedure is repeated until I at 17 is reached with cycle 2 at k = 18 at A, etc. THE M-COMPONENT GEOMET RY

0 .. 145

t

1

zt

Range variable

t.i

2

Complex number entry in terms of t

1

M 1

1

2

z

M-Component and mirror image (blue)

1.5 1 0.5

Im M t Im M t

2.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

0.5 1 1.5 Re M , Re M t t

Figure (1)

One should be aware that the first zero, which can be found in Ref (10), is alone on the second cycle. There are no zeros on the first cycle. The number of zeros per cycle increase slowly until there are an infinite number on the cycle at infinity. It will be shown that the entire listing of Ref (10) zeros can be computed and plotted per cycle. I would like to pause at this point to say a word about the descriptor that I chose to call k  2π  the RH cycle that I just introduced, viz.,   . This term is parallel to, and appears 2  ln ( 2 )  to be more suited than the ln(n), discovered by Gauss, in its relationship to the PNT and the RH (see Tables (2 and 3)). The procedure described in Fig. (1) and Fig. (2) is continued ad infinitum until the entire set of the number line values approach infinity. These number line values are the k values from whence the cycle values are generated. Neither the odd nor even integral k values can generate the zeros of the zeta function. The even integral k values generate the beginning and terminating values of each cycle, while the odd integral create the midcycle values It is clear that only the fractional values of k generate Ref (10) zeros. A zero 5

cannot occur if the “Excess of Nines = 1”. It will be explained that this behavior is also responsible for the so-called Lehmer’s phenomenon.

Figure (2)

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Graphic of the M-Component

Figure (3) If the range variables for Fig (1) and Fig (2) are made very large, the envelopes of the figures will transform to the above graphic with the following equation: M = − cos(ϑ ) + (cos(ϑ )) 2 + 1

(1.6)

2 −1 ≤ M ≤

2 + 1 . It is now easy to

which is consistent with the previously stated

see that arg (M)>90° produces an “excess of nines = 0” in column A in Fig.(2) and column B follows with the excess of 9’s =1, which always precludes the existence of a Ref.(10) zero. The remaining columns, C through I, all have k values with the potential for producing zeros. I have not seen any mathematical references to the principle of the “excess of nines”, and yet I have discovered, and will illustrate later in this document, that it is a fundamental property of both the Riemann Hypothesis and the Prime Number Theorem.

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All period cycle values were computed by multiplying by the k values as previously illustrated: k  2π  t =  (1.5) Repeated  2  ln ( 2 )  Utilizing equation (1.5), the following partial table of values was produced demonstrating the way the beginning, mid-cycle, and terminating values of the RH cycles through cycle 20 were calculated. Naturally, the list could continue to infinity. Table (1) The Period Cycles for the RH Even k Values 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Period Cycle Start and End 0 9.06472 18.12944 27.19416 36.25888 45.32360 54.38832 63.45304 72.51776 81.58248 90.64720 99.71192 108.7766 117.8414 126.9061 135.9708 145.0355 154.1002 163.1650 172.2297 181.2944

Odd k Values 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Mid-Cycle Values 4.53236 13.59708 22.66180 31.72652 40.79124 49.85596 58.92068 67.98540 77.05012 86.11484 95.17956 104.2443 113.3090 122.3737 131.4384 140.5032 149.5679 158.6326 167.6973 176.7620

Cycle Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Zero is the beginning of period cycle 1 and 9.06472 is the ending value. Clearly then, period cycle 2 extends from 9.06472 to 18.12944, with 4.53236 and 13.59708 as the midcycle value of each, respectively. The first zero, t = 14.134725 Ref (10), is the only zero on cycle 2. All of the remaining cycles have two or more zeros with the number increasing to infinity as k approaches infinity.

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III Theorem Approach to the Proof of the Riemann Hypothesis The proof of the Prime Number Theorem (PNT) was based on a conjecture by Chebyshev in 1851 that if the following expression has a limit it must be:

lim

n 

Where:

π (n) ln(n) 1 n

(1.7)

 (n)  is the number of primes up to the number n.  (n) is unrelated to the constant 3.14159. ln(n) = the natural logarithm of n to the base e = 2.71828…

Carl Friedrich Gauss, the prince of mathematicians, and Legendre among others, were n aware of the uncanny agreement of the ln(n) and , but none were successful  ( n) in its proof. In 1859, Bernhard Riemann tried to prove the conjecture by utilizing Leonhard Euler’s product formula and substituting complex numbers for real variables. Riemann was able to show that the distribution of prime numbers is closely allied to the properties of the function  ( s) defined by the series first introduced by Euler called the zeta function. After introducing complex variables into the zeta function it became known as the Riemann zeta function and had to be analytically continued to be valid in the interval 0 < s < 1. Euler’s zeta function for real numbers is valid only in the interval s > 1 .  ( s) is an analytic function with a single pole at s=1. Ref (15) shows the proof of the PNT independently discovered in 1896 by JacquesSalomon Hadamard and Charles Jean de la Vallee-Poussin. IV Analysis The discovery of a parameter of the RH that is fundamental to this theorem solution is shown below in equation (1.5). This parameter could have served as a more powerful proof of the PNT if Hadamard and Vallee-Poussin had known of equation (1.5)

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Gauss discovered in his early years by simply counting the primes up to a number n, that the number of primes contained in n is related to the logarithm of n. It is not difficult to see why this became the basis of the Chebyshev’s Conjecture shown above in equation (1.7). Extracting data from Ref (15) for the PNT and modifying it in Tables (2 and 3), we can change the form of n without changing their values, i.e., 102,104,106…etc. as follows: n  102 k and,

(1.8)

ln( n)  2k ln(10)

(1.9)

where, again, k=1,2,3….  ,or the natural number line. The parameter for the RH of equation (1.5) in all respects is akin to equation (1.9) and given by:  k   2      2   ln(2) 

(1.5) Repeated

where k has the same values as above. It will be shown below that it is important to obtain the ratio of equation (1.9) to (1.5), or: PNT  (2 k ln(10))  k   2     2   ln(2) 

RH   Then: Ratio 

PNT  1.016064488 RH

(1.10)

Naturally, this ratio, I will call it Martin’s Ratio since I discovered it, is independent of k and therefore persists throughout the entire input values of the natural number line. Equation (1.5) is a direct consequence of the manner in which the Riemann zeta function was analytically continued to be valid in the interval 0<s<1 as shown in Appendix (A). The data contained in the tables below in all accounts is consistent with the data extracted from Ref (15).

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V The Prime Number Theorem (PNT) The following data was extracted directly from Ref (15) and formatted to facilitate the comparison of the RH with the PNT. n

π ( n)

n π ( n)

ln ( n )

k

102 104 106 108 1010 1012 1014 1023

25 1, 229 78, 498 5,761, 455 455,052,511 37,607,912,018 3, 204,941,750,802 1,925,320,391, 606,818, 006, 727

4.000 8.137 12.739 17.357 21.975 26.590 31.202 51.939

4.605 9.210 13.816 18.421 23.026 27.631 32.236 52.959

1 2 3 4 5 6 7 11.5

2k ln ( 10 )

k 2π 2 ln ( 2 )

4.605 9.210 13.816 18.421 23.026 27.631 32.236 52.959

4.532 9.065 13.597 18.129 22.662 27.194 31.727 52.122

Table (2)

π ( n) ⋅ 2k ln ( 10 ) n 1.151293 1.131951 1.084490 1.061299 1.047797 1.039145 1.033151 1.019639

π ( n ) k  2π ⋅  n 2  ln ( 2 ) 1.133090 1.114054 1.067344 1.044520 1.031231 1.022716 1.016817 1.003518

  

Table (3) The first column of Table (3) indicates the tendency of the calculations to verify Chebyshev’s Conjecture; the values are decreasing and approaching unity as the k values approach infinity. Fortunately, I do not have to prove this as Hadamard and ValleePoussin have already done so, Ref (15). The second column of Table (3) was constructed using the RH equation (1.5) and shows the definite link between the RH and the PNT. In fact, it could be said that the RH values are better suited for the proof of the PNT since the approach to unity is faster. Further, it would be illogical to assume the values from the RH are approaching some limit other

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than unity, especially since the constant difference between the two columns can be reconciled by multiplying the RH values by the previously determined Martin’s Ratio =1.016064488 to obtain the values in column 1, or vice-versa. The fourth and sixth columns of Table (2) show the logarithm of n and its remarkable agreement with column three discovered by Gauss. Column seven, a better agreement with column three, was discovered accidentally by the author from the analytical continuation of the Riemann zeta function (note the hiatus between n =1014 and 1023), or:

ζ ( s) 

 ( 1) 1  1 s  1 2 ns n 1

n 1

(the change of variable is not relevant)

(1.3) Repeated

The term multiplying the summand is alone responsible for the values in column seven. Gauss could not explain why the number n divided by the number of primes up to n were nearly equal to ln(n). I believe the reason for the behavior of the Gauss parameter and for the parameter I discovered are both related to a principle of “the excess of nines” (Ref (2). This principle and the reason why I think it links the RH to the PNT will be described later. VI The Limit Process applied to the Chebyshev Conjecture.

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VII Principle of Excess of Nines.

Since the value of s in equation (1.3) depends on the complex number s 

1  i t 2

 2   k   2  where t  Cycle       and k=1,2,3....n of the natural number line.  ln(2)   2   ln(2)  It is evident that all odd or even (whole number) values of k produce an excess of nines=1 and therefore cannot generate a zeta function zero. Further, it is true that all zeros occur from the fractional values of the k’s; when an unscheduled “Excess of Nines =1” occurs, 13

so does a “Lehmer event”. The plotted data briefly passes through an excess of nines =1, wavers, then becomes fractional again, crosses the real axis and produces a zero. These events are cyclic, happening many times along the path to infinity (see Appendix(A)).

Figure (4)

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Figure (5) From Fig.(5), it is clear that the parameters of the PNT are greater than the RH by the constant Ratio =1.016064488 throughout the natural number line. This Ratio can be used to reconcile any difference between the two, e.g., the first Lehmer event for the PNT 17143.8218  1861.367 instead of 1891.2689 where it occurs for the RH, occurs at 2 2 ln(10)

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1891.2689  1861.367 . Appendix (A) contains several plots that illustrate other 1.016064488 Lehmer events. or

2 1891.2689   17143.8218 ln(2)

Figure (6) 16

In the research I have done on the RH, it has been interesting to notice that it seems to be that more people are trying to disprove the conjecture than to prove it. In Ref (14), a crude graph is presented that purports to indicate a possible failure of the RH. The graphic plots the gradient of the hills and valleys along Riemann’s “critical line”. The point cited as a potential failure appears to occur at the so-called “Lehmer’s 2  17143.8218 , the precise value as in Phenomenon” (which it isn’t) at t  Cycle  ln(2) my calculations and in the Ref (10) compendium of zeros of the zeta function. There are many of these events that occur cyclically which I have renamed “Lehmer’s Events” as described in Appendix (A). These events are responsible for the cause celebre that the RH might be false which it is not.

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VIII Theorem Statement The Riemann Hypothesis: The non-trivial zeros of the Riemann zeta function have 1 1 the complex input s   i t , where the real part e( s )  and the imaginary part 2 2 ( s)  i t . The parameter t can have input values from the RH or the PNT to replicate the Ref (10) zeros as will be shown in Appendix (A). The value from the RH is:

 k   2  .   2   ln(2) 

t 

The value from the PNT is: t  2 k ln(10) . The k value in either expression is the natural number line k  1, 2,3.... . It has already been shown that the constant ratio of parameters is: Martin’s Ratio

PNT  1.016064488 RH

which states that PNT values can always be found from the RH values and vice versa. This fortunate set of circumstances, coupled with the “excess of nines principle”, n provides a glimpse of understanding into the close relationship of ln(n) with  ( n)  k   2  discovered by Gauss and   discovered by the author. Either value has been  2   ln(2)  shown to satisfy Chebyshev’s Conjecture, with the RH value tending to provide a faster progression to unity as k   . The proof of the PNT has been established. Therefore, the Prime Number Theorem proof is a corollary to the proof of The Riemann Hypothesis. The proof is conclusive that all the non-trivial zeros (primes) are on the “critical line” at x = ½ and that they   as t   (equation (1.3). This completes the proof of the Riemann Hypothesis verifying that it is, indeed, true.

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IX Summary Comments (1) The discovery of the RH period cycle was key to the solution of the RH. (2)The uncanny agreement of the RH period cycle, discovered by the author, with the PNT period cycle (discovered by Gauss), demonstrates the immutable link between the PNT( proved in 1896) and the RH. This solid association of the two is at the heart of the Riemann Hypothesis. The RH period cycle and its association with the PNT has been fully explained as the result of the principle of the “Excess of Nines.” (3) This solution would not have been possible without the Ref (10) list of zero’s as a “sanity check”. . (4) Lehmer’s Phenomenon is explained as a consequence of the manner in which the M-Component generates the period cycle and frequency of the RH and is not a phenomenon threatening the possibility of the truth of the RH in any way. (5)The reason the graph of the gradient of the hills and valleys shown on page 217 of Ref. (14) does not cross the abscissa is also a consequence of the M-Component. (6)An explanation of the frequency of the RH as multiples of 144, i.e., 288, 432, 576, etc., to infinity was completely defined and shown to be due to the “Excess of Nines” (7) The Riemann Hypothesis was shown to be true, from a corollary proof by the Prime Number Theorem. The parameters from the RH or the PNT can be interchanged to replicate the zeros of Ref (10). In addition, the RH and the PNT were both shown to satisfy Chebyshev’s Conjecture. (8 The values of columns 6 and 7 in Table (2), illustrated for the Prime Number Theorem, can be completely interchanged to produce the zeros of Ref (10) as shown in Appendix (A). This single action is sufficient to verify the truth of the RH if the PNT proof is valid. (9) I made the assertion (Section VII), that all odd and even (whole number) k values of the natural number line produce an “excess of nines”=1, and therefore cannot be an RH zero (prime), nor a PNT (prime). The even numbered k values were shown to be responsible for the beginning and terminus of each cycle to infinity. A consequence of this is only fractional values of k can generate zeros or primes, and the final cycle at infinity must end with an even value of k . If one believes as I do, it would be an incongruence of Nature to arrive at infinity with an incomplete cycle. Therefore, as Bombieri opined in Ref (13) concerning the prime numbers, “There is no prime at infinity – that’s clear to me at least”. My last comment supports Bombieri’s contention.

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X References: (1) Edwards, H. M. “Riemann’s Zeta Function”, Dover Publications, Inc. Mineola, New York. (2) James, G. and James, R. C. "Mathematics Dictionary" Princeton, New Jersey: D. Van Nostrand Company, Inc. (3) Gullberg, J. "Mathematics From the Birth of Numbers" New York, London: W. W. Norton & Company. (4) Wikipedia, Legendre’s Constant (http://en.wikipedia.org/wiki/Legendre’s_constant) (5) Patterson, S. J. "An Introduction to the Theory of the Riemann Zeta-Function" New York: Press Syndicate of the University of Cambridge. (6) Ogilvy, C. S. and Anderson, J. T. "Excursions in Number Theory" New York: Dover Publications, Inc. (7) Courant, R. and John, F. "Introduction to Calculus and Analysis,Vol. I and II" New York, London: Interscience Publishers, A Division of John Wiley and Sons, Inc. (8) Singh, S. "Fermat's Enigma" New York: Anchor Books, A Division of Random House, Inc. (9) Mathcad Software, MathSoft, Inc., Cambridge, Massachusetts, 02142. (10) Odlyzko, A. "Andrew Odlyzko: Tables of Zeros of the Riemann Zeta Function." (11) Download from the Claymath URL: Problems of the Millennium:The Riemann Hypothesis, E. Bombieri. (12) Gradshteyn I.S. and Ryzhik I.M., “Table of Integrals, Series, and Products”: Academic Press (Sixth Edition). (13) Sabbagh, K. “THE RIEMANN HYPOTHESIS, The Greatest Unsolved Problem in Mathematics” Farrar, Straus and Giroux / New York. (14) Du Sautoy, M. “The Music of the Primes” Perennial (An Imprint of Harper Collins Publishers) (15) “Analytic Number Theory: The Prime Number Theorem”, Britannica CD, Version 99 C 1994-1998.

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XI Appendix (A) The data contained in this appendix is not necessary to substantiate the proof of the Riemann Hypothesis but there is much evidence here to support it. The information presented in this document is sufficient to prove the Riemann Hypothesis is true, except possibly for the few that have the temerity to deny it without proof to the contrary. Lehmer’s Phenomenon I will show the indisputable correlation of the characteristics of the Riemann Hypothesis with those of the Prime Number Theorem. They both will replicate the zeros of the zeta function shown in Ref (10), utilizing the input data from either, and will differ only by the cycle of occurrence that I have defined. I have devoted much attention to a so-called “Lehmer’s Phenomenon (LP)”, which I would have been content to not mention further; however, Ref (14), (pgs. 213-219), has devoted about as much time on the subject without recognizing that it’s the same debutant in different attire. The designation is an over-statement and I have elected to rename it “Lehmer’s Events (LE)”. There seems to be a prevailing notion among some mathematicians that the LP may be the signal of a counter-example to the RH making it false. The LP occurs at t = 17143.3218 (a computed value also verified by Ref (10)), that correlates to the precise point shown in Ref (14). I will defer the reader to the cited reference, and present my reasons for declining the lofty name. I was aware of the LP long before it appeared in Ref (14), but was unaware of its influence in the possible conclusion that it might be a “prophet” to the falsity of the RH. This event occurs because of the mechanization of the M-Component described 2  9.06472028 will increase by a value of 1 when previously. Recall the constant ln(2) multiplied by each group of 16 cycles, i.e., 16, 32, 48, etc. Since the fractional portion of the constant is (0.06472028)(16)=1.03552448, a spurious introduction of an additional increase of 1 causes an unscheduled “excess of nines=1”. This results in a momentary “waver” in the plot (computed data), due to the fact that a zero cannot occur at an excess of nines =1 (See Fig (4)). A sudden return to fractional values of t permits the plot to cross the real axis, thus creating a “Lehmer Event” (LE). An LE will occur at each increment of 144 units of the natural number line, or (9)(16)= 144 as illustrated in Fig (2 and 4). I have defined this interval as the frequency in accordance with Ref (2): The frequency of a periodic function in a given interval is the quotient of the length of the interval and the period of the function (i.e., the number of times the function repeats itself in the given interval). It should be obvious, then, that the first frequency occurs at natural number 32 or k= 32 which by our definition of cycles is 16 equating to the first frequency interval of 144.The following sequence of natural numbers corresponding to the second, third, and fourth frequencies, etc., is 64, 96, 128, 160….  , giving cycles 32, 48, 64, 80…  , with

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corresponding frequency intervals of 145.035, 290.071, 435.106, 580.142, 725.177…  . The list of nominal frequencies, then, is144, 288, 432, 576…  . Using this procedure, it is clear that the LE (computed and extracted from the Ref (10) list as t = 17143.8218) is between cycles 1891 and 1892 as will be shown, and therefore is in a region where crossing the real axis must occur, thus precluding a counter-example for the RH. This potential occurrence repeats over and over at each frequency generating additional LE examples. The occasional reluctance of the graph shown in Ref (14) to cross the real axis is a direct result of these LE’s. The first four LE frequency intervals are cycles 1892, 3784, 5676, and 7568 which have been computed and plotted. The method was developed by the author that obviates the necessity for using the Riemann-Siegel formula for finding the RH zeros. The LE plots for the mentioned cycles have the following t values, computed and extracted from Ref (10), they are: First Event………..17143.8218 Second Event……..34295.3720 Third Event……….51448.0763 Fourth Event………68597.9714 These values have a nearly constant difference, substantiating the appearance of an LE at the specified cycles and frequencies. The procedure that I devised for finding any of the LE was generalized from the method from which I found the first one. Recalling that each group of 16 cycles produced the nominal frequencies of (9) (16) = 144. Using this nominal, the number of groups of 16 cycles is needed starting with the known LP = 17143.8218 / 9.0647202 =1891.26859. It was evident from this, that the cycle beginning was probably 1891 and the ending 1892. Since cycle numbering was chosen as the ending number the LP must be on cycle 1892. Now, the value of cycle at 1892 / 144 = 13.13888889 (the number of 16 cycle groups). The actual frequency interval for the first group of 16 was 145.0355245; thus the frequency for cycle 1892 must be (145.0355245) (13.13888889) = 1905.605642. Finally, (1905.605642) (9) = 17150.4508. This is the precise value of the terminus of this cycle enclosing the first LP. The second LP cycle will be twice the first, or 3784. This procedure can be repeated to obtain any LE desired. The first LE =17143.8218 has been computed and plotted below as Figure (7). The complete procedure is presented in this first data sheet which will not be included for the plotted Fig (8) .If the reader desires to check the plotted data it will be necessary to procure a copy of Ref (9), Version 8 or later.

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23

2 1891.2689   17143.8218 ln(2)

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22 3783.39 3783.39    34295.372  34295.372 ln(2) ln(2) 1861.3668 (2 2 ln(10))

The plots for the third and fourth LE, 51448.0763 and 68597.9714, respectively, would follow the same procedure as Fig. (7 and 8), so these two LE will not be plotted. It is important to point out, again, that all the zeros of Ref (10) can be replicated using either the RH values of t or the PNT values. The k values used are the same for either, and both satisfy the Chebyshev Conjecture (Section VI); however the plotted data will, for a given RH cycle, appear earlier for the same data appearing on the equivalent PNT cycle. To illustrate this, the first LE plot will be shown next after locating the PNT cycle exactly; then, producing the plot using PNT values exclusively, except for cycle number:

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26

1861.3668 (2 2 ln(10))

For the reviewers that are still unconvinced after confronting the indisputable evidence of the arguments presented in this document that the Riemann Hypothesis is unequivocally true, I will, using Chebyshev’s Conjecture (proven), administer the coup de grace. Recalling equation (1.7): lim

x 

 (n) ln(n) 1 n

(1.7) Repeated

and utilizing this statement of Chebyshev and the software of Ref (9), and emphasizing that this should not be deemed a computer solution, I submit the following additional Table (4). This table is a comparison of RH and PNT parameters and values and will be used in the mathematical exercise that follows:

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Comparison of some pertinent values and parameters of RH and PNT RH k

n  102 k

1

10

2

104

3

106

4

108

5

1010

6

1012

7

1014

8

1016

9

1018

10

1020

89

10178

90

10180

3783

107566

3784

107568

Cycle 

k 2

tb

tm 4.532

2

1

0

PNT te

tb

9.064

0

13.597

2

9.064

18.129

18.129

18.421

36.259

36.259

27.631

45.324

398.848

36.841

46.052 409.860

407.912 405.255 17145.9

1892

36.841 41.447

403.380

45

27.631 32.236

40.791 5

18.421 23.026

27.194

27.194

9.210

9.210

31.727 4

17141.4

te

13.816

22.662

3

tm 4.605

414.465 17421.4

17150.4 17416.7

17425.9

Table (4)

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 (n) ln(n)  1 . Using appropriate values from Ref (4), or: n  n

The conjecture states the lim

 (n)  1925320391606818006727 n  1023 ln(n)  23 ln(10)  52.9594571  (n) ln(n)  1.019639 n Indications are that the PNT value is very near Martin’s Ratio at n   which results in the RH value of Table (3) attaining the value 1 prior to the PNT value as was suggested earlier and in Appendix (B). Chebyshev’s Conjecture appears to be approaching the indicated limit as n approaches infinity. Note that all of these computations are confined to the PNT. The technique for arriving at the RH values can be derived by a judicious application of Martin’s Ratio equation (1.10) to the PNT values and parameters as illustrated previously. It will be shown in Section XII, Appendix (B) that Legendre’s Constant instead of being 1.08366 might be equal to Martin’s Ratio (1.016064488). One Final Note on Lehmer’s prediction that his phenomenon (event) would recur infinitely often, reported Ref (1), 8.3, pg.179, is true for the reasons given at the beginning of this Appendix. Ostensibly, from data supported by the Gram point methodology, a pair of zeros in the vicinity of the 13,400,000 th zero were separated by a distance inconsistent with Gram point data in this region. The author of Ref (1) remarked that it would be interesting to have a graph, such as Lehmer’s (Fig.3) on the facing page, in this region. The following explanation and graphs will not only provide such a graph in terms of t, it will corroborate the procedure at the t=13,400,006.8 Lehmer event shown on pg.22 of this document, thus vindicating the prediction of this author, and simultaneously vindicating the prediction of Lehmer. Following the procedure at the beginning of Appendix (A) for the first Lehmer event, the one occurring near the 13,400,000th zero will be determined. The first frequency interval occurs at the 16th Cycle at (9)(16)=144 units of the natural number line as explained in the body of this document. Each succeeding frequency interval is a multiple of the frequency at the 16th Cycle. It is then only necessary to determine the number of groups of 16 Cycles contained in the Cycle for the RH at the desired zero. From the RH data sheet this can be observed to be 10265.69444. At the first frequency, it was explained that the actual value is (9.064720284)(16)=145.0355245. This suggests that the frequency near the 13,400,000th zero is (10265.69444)(144)=1478259.99. When this latter value is multiplied by 9.064720284, the terminal value, Te =13400013.4 is given for the RH (Fig. (10).This same terminal value Te exists for the PNT (Fig. (11) but the Cycle value of the PNT must be decreased by Martin’s Ratio to 1454888 due to the earlier occurrence

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of the PNT with larger values than the RH. The beginning value Tb = 13400004.34 for the PNT is less for the same reason. The following three figures replicate the same event in three different ways. Figure (10) is constructed exactly like the first Lehmer Event in Fig.(7) above, except the much larger Cycle value =1478260 was substituted after finding this value by a ratio of similar values of the first Lehmer Event to the desired event, or: 1892 13400000  1478258.52 (rounded to 1478260). 17150.4508 Referencing Fig.(10), it can be observed that the RH values were utilized to produce the graphic. Fig. (11) uses the PNT values and Fig. (12) applies Martin’s Ratio to the RH values to obtain the equivalent PNT values. All of the figures produce the exact same plots verifying the cyclic nature of the Lehmer Events. For all three plots, the calculated value of the Lehmer Event occurs approximately at the value 13400006.8 atCycle = 1478260 for the RH and 1454888 for the PNT.I am not a believer in the Gram point process so I will let the reader associate those values with the plotted data.

A personal note from the author: It has been an unbelievable journey chasing the RH for nearly seven years in almost constant pursuit. I am certain, although others may disagree, that the solution is correct. A few of the convoluted paths taken, all of which were arduous and some times misleading, led me to discover things that I believe are unique and key to the solution. Much of the information posted on the web about the RH is outright false, and some is only confusing. My discovery of the RH cycle parameter is in all respects the equivalent of Gauss’ discovery of the relationship of the ln(n) to the PRT, although both were serendipitously found. Also Martin’s Ratio between the PNT and RH is a constant that allows many values of either to be interchanged. In addition, the “Excess of Nines” principle is a fundamental attribute of both. I undertook an attempt at solving the RH because of the faith a friend had in me who knew of my life-long love of the subject of mathematics; and not because of the bounty placed upon its solution. Riemann was unquestionably a genius. What else could you call a mathematician that has confounded other mathematicians for a century and a half? Mack L. Martin July10, 2007 (www.riemannsolved.com)

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31

32

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XII Appendix (B) In Ref (4), Adrien-Marie Legendre’s conjecture that the prime counting function: n  ( n)  ln(n)  A(n) lim  A(n)   1.08366...

Where

n 

The quantity 1.08366…was called Legendre’s constant, and in Ref (4) is stated as a “phantom” that really does not exist, in fact, Ref (4) states the best limit value of A(n) turns out to be 1. Thus, there is no such constant. Later, Carl Friedrich Gauss examined the numerical evidence and concluded that the limit might be lower. The purpose of the following exercise is to show, once again, that Gauss was more of an intellect than most of his contemporaries, and acceded that the limit might be lower but did not specify, however, that the limit would be 1. As it turns out the limit is lower than 1.08366 but not 1, and from all my previous work the value should have been predictable as will now be shown: Re-arranging the equation at the top of this page to: lim A(n)  lim(ln(n)  n 

n 

and lim A(n)  lim( n 

n 

n )  ( n)

k 2 n  ) 2 ln(2)  (n)

for the PNT

for the RH

(1.11)

(1.12)

And substituting the appropriate values from Table (2) on page 11 for each of the columns in both the below Tables (5 and 6), including the values determined by Helge von Koch in 1901 (Ref (4)):

 (n)  1,925,320,391, 606,818, 006, 727 n=1023 and n  51.939 for both the PNT and RH.  ( n) ln(n)  52.959 And for equivalent of the ln(n), i.e.,

for the PNT. k 2   52.122 2 ln(2)

for the RH.

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Table (5) - Values for the PNT

k

n  102 k

ln(n)

1 2 3 4 5 6 7

102 104 106 108 1010 1012 1014

4.605 9.210 13.816 18.421 23.026 27.631 32.236

4.000 8.137 12.739 17.357 21.975 26.590 31.202

0.605 1.073 1.077 1.064 1.051 1.041 1.034

0.605 1.073 1.077 1.064 1.051 1.041 1.034

11.5

1023

52.959

51.939

1.02005

1.02005

n  ( n)

(ln(n) 

n )  ( n)

A(n)

Table (6) – Values for the RH

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n  102 k

k 2 n ( ) 2 ln(2)  ( n)

(

1 2 3 4 5 6 7

102 104 106 108 1010 1012 1014

4.532 9.065 13.597 18.129 22.662 27.194 31.727

4.000 8.137 12.739 17.357 21.975 26.590 31.202

0.532 0.928 0.858 0.772 0.687 0.604 0.525

0.532 0.928 0.858 0.772 0.687 0.604 0.525

11.5

1023

52.122

51.939

0.183

0.183

k

k 2 n  ) 2 ln(2)  (n)

A( n)

A(n) instead of The last entry in both tables (5 and 6) should have been obtained at lim n  1023 which is clearly an impossible quest. Even so, using the Helge von Koch data (~ 1.02005 from the table (5) is sufficient to show Legendre’s constant (~1.08366…) is too large. Gauss agreed that Legendre’s constant might be lower. Ref (4) states that the best limit for A(n) is 1 and there is no such constant existent as claimed by Legendre. From the research I have performed, I agree the value of Legendre’s constant is too large, but it does exist and it is not equal to 1 for the PNT. I plotted the values in the last column of Tables (5 and 6) above against the k values and the result was surprising. Each trace increases from k = 1, both breaking to a downward slope at exactly 2, but not with the same decreasing slope. The RH plot slope decreases rapidly to a value of 0.183 as k values increase to infinity. This indicates that the difference: k 2 n ( )( ) ) 2 ln(2)  (n) between the RH value and the ratio of the number n divided by the actual number of primes contained in n is zero at infinity, and this implies the RH value is 1 at infinity which is consistent with Chebyshev’s Conjecture. On the other hand, the difference between the ln(n) and Gauss’ ratio of the number n to the number of primes @ n appears to remain at a near constant value. It is very suggestive that the PNT plot is approaching the value which I named previously as Martin’s Ratio = 1.016064488. This is indeed less than Legendre’s Constant but still not equal to 1. Table (3) clearly shows a tendency for this to occur. 36

37

From Fig. (13) and information contained in Tables (2), (3), (5) and (6) developed from various referenced sources, it can be concluded that the PNT appears to be approaching the value defined as Martin’s Ratio = 1.016064488 vice Legendre’s Constant 1.08366 and the RH equivalent value appears to be approaching 0 asymptotically as k (and n) approach infinity; both satisfy Chebyshev’s Conjecture. The relationship of the RH to the PNT will be illustrated in this final summary. From Ref (4), the largest values obtainable for  (n) =1,925,320,391,606,818,006,727 for n = 1023 . Since I have defined n = 102k , k = 11.5 and the values from Table (5) that should be substituted in equation (1.11) are: lim A(n)   52.959  51.939  1.02005 k 

and in equation (1.12) from Table (6): lim A(n)   52.122  51.939  0.183 k 

Table (5) already shows Legendre’s Constant =1.08366 is too large and by extrapolating the data in Table (6) and Fig.(13), the value of n will probably be large since the approach to 0 on the abscissa is asymptotic. Nonetheless, the Fig.(13) blue trace is approaching zero, indicating no error between the calculated and actual number of zeros, thus assuring that Chebyshev’s Conjecture is approaching 1 for the RH.

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The red trace of Fig.(13) and Table (5) stabilizes at a value slightly >1 and it is probable that it is approaching Martin’s Ratio for the PNT, then:

 (n) (2k ln(10))  1.016064488 , and as shown previously, dividing n both sides of this equation by Martin’s Ratio confirm the Chebyshev Conjecture for the RH, or:

 (n) 2k ln(10)  (n)  k   2       1 n 1.016064488 n  2   ln(2)  This completes the proof of the Riemann Hypothesis, verifying that it is indeed true.

Mack L. Martin PE (State of Ohio) 345 Gray Road Melbourne, FL 32904 E-mail: [email protected]

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