A General Characteristic Equation For Zeta Function

by

Huen Y.K.

CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg
(A Short Communication: First release: 10/7/98, Revised 12/7/98)


Abstract

Riemann's Hypothesis is generally regarded as the most important unsolved mathematical problem today. All real parts of the non-trivial zeros of zeta are supposed to be exactly 1/2. Nobody knows for certain if this is true but numerical evidence seems to suggest that it is. Attempts to predict the roots of zeta accurately have only met with partial success. In 1986 it was shown that the first 1,500,000,001 nontrivial zeros of the Riemann zeta function do indeed have real part one-half [1]. Since tables of roots of zeta are easily available, in this paper the author applies reverse engineering to synthesize a general characteristic equation which can predict the roots of zeta to any degree of approximations depending on the accuracies of the roots being used to develop it [3]. This exercise provides considerable insights on why Riemann's Hypothesis is so difficult to prove. It is because there is no shortcut formulation which could replace the general characteristic equation. In fact any shortcut will always introduce errors in predictions since it would be extremely difficult to find a shortcut which is isomorphic to the general characteristic equation.


1. Introduction

Riemann's Hypothesis remains one of the most important unsolved problem in this centuary. So far all predictions of roots of zeta have only been partially successful probably indicating that we are nowhere near uncovering its fundamental principle. Instead of attempting to predict analytically the roots of zeta from Euler's zeta function whether in the summative form or the product form, the author has applied reverse engineering to derive the general characteristic equation which theoretically could predict all these roots accurately. The general formulation turns out to be very lengthy and computationally disadvantageous because the coefficients grow rapidly to very large magnitudes. Nevertheless, the general characteristic equation can be generated by a symbolic package and provides considerable insights on why Riemann's Hypothesis is so difficult to prove analytically.

Characteristic equations are familiar to those working with sampled data control systems using z-transforms where the poles of this equation could be used to predict the stability of a control system in the z-plane [2]. Since the original zeta function is defined only in the open form, it is difficult to predict the roots anayltically. Here, the author proposes to take the path of reverse engineering to derive its general characteristic equation. This is not too difficult since computed roots of zeta to many digits of accuracies are available from Odlyzko [3].

Originally, the author has proposed intuitively a simple version of characteristics equation as given by equation (1). This formulation will yield both complex and conjugate roots including the real part of ± ½.

One advantage of this equation is that the roots will always occur in conjugate pairs with the real part of 1/2. Table 1 shows some sampled root extractions at various chosen ranges where it can be seen that even the first root of zeta can be predicted with good accuracy. Predicting large values of roots of zeta is not a problem since the accuracies get better with increasing magnitudes of r.

Table 1 - Roots From Characteristic Equation Given by Equation (1)
zroots:sum(z^roots(s^2-s+(r+1/2)),r,195,205);

(ii) The first 10 positive complex roots (r from 195 to 205):
[11 * z^s = z^(14.3265487818944d0 * %i
+ 0.5d0) + z^(14.2916059279564d0 * %i + 0.5d0)
+ z^(14.2565774293832d0 * %i + 0.5d0)
+ z^(14.2214626533278d0 * %i + 0.5d0)
+ z^(14.186260959111d0 * %i + 0.5d0)
+ z^(14.1509716980848d0 * %i + 0.5d0)===actual roots: 14.1347251417347
+ z^(14.115594213493d0 * %i + 0.5d0)
+ z^(14.0801278403286d0 * %i + 0.5d0)
+ z^(14.044571905188d0 * %i + 0.5d0)
+ z^(14.0089257261218d0 * %i + 0.5d0)
+ z^(13.973188612482d0 * %i + 0.5d0),

zroots:sum(z^roots(s^2-s+(r+1/2)),r,440,450);

(iii) The first 10 positive complex roots (r from 440 to 450):
[11 * z^s = z^(21.2190951739228d0 * %i + 0.5d0)
+ z^(21.1955183942266d0 * %i + 0.5d0)
+ z^(21.17191535974d0 * %i + 0.5d0)
+ z^(21.1482859825566d0 * %i + 0.5d0)
+ z^(21.1246301742776d0 * %i + 0.5d0)
+ z^(21.1009478460092d0 * %i + 0.5d0)
+ z^(21.077238908358d0 * %i + 0.5d0)
+ z^(21.0535032714272d0 * %i + 0.5d0)
+ z^(21.029740844813d0 * %i + 0.5d0)
+ z^(21.0059515376d0 * %i + 0.5d0)===actual root: 21.0220396387715
+ z^(20.9821352583572d0 * %i + 0.5d0),

sum(z^roots(s^2-s+(r+1/2)),r,50175,50180);

(iv) The first 10 positive complex roots (r from 50175 to 50180):
roots1 = z^(224.009486406268d0 * %i + 0.5d0)
+ z^(224.007254346818d0 * %i + 0.5d0)===actual root: 224.007000254604
+ z^(224.005022265126d0 * %i + 0.5d0)
+ z^(224.002790161194d0 * %i + 0.5d0)
+ z^(224.000558035019d0 * %i + 0.5d0)
+ z^(223.9983258866d0 * %i + 0.5d0),




2. General Characteristic Equation

The global general characteristic equation of zeta function can only be written down if we know all the roots of zeta beforehand. The situation is somewhat akin to the product version of zeta function where we can only find the roots if the primes are known. Let us define the global list of roots of zeta as given by equation (2):

Using indexed notation for k1,k2,..., from equation (2), the general characteristic equation can be written down as follows:


               /    ub               \
               | --------'           |
               |'  |  |              |
           1 + |   |  |    (z - k(i))| = 0  .........(3).
               |   |  |              |
               |   |  |              |
               \  i = 1              /
Equation (3) looks deceptively simple but the global statement would entail taking in an infinite set of zeta roots from equation (2). Although the expanded version of equation (3) is still univariate in z, roots extraction of such a large system would strain the resources of the most powerful of computers. Strictly the equation is more useful for gaining algebraic insight than for intensive numerical computations.

Equation (4) shows finite expansions with only the first five roots of zeta:

(1+expand(product((z-f(k)),k,1,5)));

z^5

-z^4 * ( f(5) + f(4) + f(3) + f(2) + f(1) )

z^3 * ( f(4) * f(5) + f(3) * f(5) + f(2) * f(5) + f(1) * f(5) + f(3) * f(4) + f(2) * f(4)
+ f(1) * f(4) + f(2) * f(3) + f(1) * f(3) + f(1) * f(2) )

- z^2 * ( f(3) * f(4) * f(5) + f(2) * f(4) * f(5) + f(1) * f(4) * f(5) + f(2) * f(3) * f(5)
+ f(1) * f(3) * f(5) + f(1) * f(2) * f(5) + f(2) * f(3) * f(4) + f(1) * f(3) * f(4)
+ f(1) * f(2) * f(4) + f(1) * f(2) * f(3) )

z * ( f(2) * f(3) * f(4) * f(5) + f(1) * f(3) * f(4) * f(5) + f(1)* f(2) * f(4) * f(5) + f(1) * f(2) * f(3) * f(5) + f(1) * f(2) * f(3) * f(4) )

- f(1) * f(2) * f(3) * f(4) * f(5)

+ 1 = 0 .........................................................(4).

Given below is a real numerical example using rounded roots given by zetaroots = {14,21,25,30,32}. Both the unexpanded version in equation (6) and the expanded version in equation (7) give the same results. It can be observed that even with five zeta roots, the polynomial is raised to the 5th power and in general with n roots, it is raised to the nth power. One saving grace is that the equation still yields correct predictions of roots by using a finite number of zeta roots. It should be correct from n = 2 all the way to infinity provided your computer is capable of handling multiprecision arithmetic.

1+expand((z-14)*(z-21)*(z-25)*(z-30)*(z-32));

z^5 - 122 * z^4 + 5849 * z^3 - 137428 * z^2 + 1577940 * z - 7055999 = 0.....(5)

roots(z^5 - 122 * z^4 + 5849 * z^3 - 137428 * z^2 + 1577940 * z - 7055999);...(6).

[z = 13.9999549069192d0,
z = 21.0003607906146d0,
z = 24.9993506501342d0,
z = 30.0006945054784d0,
z = 31.9996391468532d0]

roots(1+expand((z-14)*(z-21)*(z-25)*(z-30)*(z-32)));.....(7).

[z = 13.9999549069192d0,
z = 21.0003607906146d0,
z = 24.9993506501342d0,
z = 30.0006945054784d0,
z = 31.9996391468532d0]



3. There is no shortcut formulation

Since the general characteristic equation given by equation (3) is the most complete statement of the roots of zeta function, one wonders whether it is possible to find a substitute shortcut formulation which will give the same performance. Equation (1) is just such a statement which only approximates the full equation by a quadratic equation. It can still predict all the zeta roots as a subset but it has lost the power of differentiating between zeta roots and nonzeta roots. This is not isomorphic mapping since many more nonzeta roots are generated in addition to the zeta roots. You will find that in order to ensure isomorphic mapping, the r variable will have to be replaced by an expression which will be as lengthy as the original general characteristic equation. Anything short of that will result in inaccurate prediction. Isomorphism is discussed extensively in refernce [4] and will not be elaborated here. All isomorphism theorem says is that two expressions are isomorphic if mappings from one to the other is 1-1 and onto. There is no shortcut here. This explains why Riemann's Hypothesis is so difficult to prove via the zeta function. This is because the roots of zeta cannot be predict analytically via the zeta function. Therefore zeta function is not isomorphic to the general characteristic equation. If roots can only be computed numerically from the zeta function, it is not in a suitable form for algebraic prediction. That is why the general characteristic equation is developed. Since this equation can predict all the roots of zeta deterministically, it must be nearer to fundamental principle connected with Riemann's Hypothesis than the zeta function.


4. Conclusions

Although the general characteristic equation given by equation (3) looks deceptively simple, it is in fact a very massive algebraic structure with very large numerical coefficients. But it represents the most accurate equation for the prediction of roots of zeta and the prediction is entirely analytical. This points to it that the formulation must be much nearer to fundamental principle than other formulations including the zeta function.

Any attempt to find a concise formulation which could replace the general characteristic equation accurately must have to be isomorphic to it, i.e., the mappings must be 1-1 and onto. This is an impossibility unless the so-called shortcut formualtion is as massive as the original one. One cannot even map the general characteristic equation onto the original zeta function nor its product version because the two latter formulations do not yield to analytical root extractions and are therefore not isomorphic to the former. This being the case, attempts to prove Riemann's Hypothesis directly through the zeta function might be futile. On the other hand, it is a paradox to prove something by supplying the information to the general characteristic equation first since we must have the roots in the first place. This is not the objective in developing the general characteristic equation but it is used to prove the difficulty of proving Riemann's Hypothesis. It is very much like the product version of zeta function which contributes to the prime number theorem.


5. Epilog

All papers posted in this website are based on the theory of sequence algebra. This is currently the only website which provides access to this theory. Readers who are not familiar with sequence algebra should read an appropriate selection from amongst the 70 odd papers posted in this website or to refer to published papers {5 to 11]. All papers posted in this website are downloadable in entirety and readable using IE or Netscape browsers from version 3 onward. Only basic HTML syntices are used in typsetting and with the exception of one paper, none uses gif files or TEX for equations. However all these papers are WYSIWYG on the browsers but not WYSIWYP (= what you see is what you print).


5. References

1. Chris. K. Caldwell: Riemann Hypothesis - Another primepage by Chris K. Caldwell.

2. Lyben W.L.: Process Modeling, Simulation And Control For Chemical Engineers, second edition, McGraw-Hill International Editions, Chemical Engineering Series, 1989, Singapore, pp 626.

3. Andrew Odlyzko: Tables of zeros of the zeta function 4. Silver C.L.: From Symbolic Logic ... to Mathematical Logic. Wm. C. Brown Publishers, Dubuque, 1994, page 290.

Huen Y.K.:The Two Alternative Versions Of Zeta Function - are these exactly equivalent?, Vol.2, paper 24, a short communication posted in
http://web.singnet.com.sg/~activweb/

5. Huen Y.K.: A Matrix Map for Prime and Non-prime Numbers, INT. J. Math. Educ. Sci. Technol., 1994, VOL. 25, NO.6, pp 913-920.

6. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ. Sci. Technol., 1996, VOL.27, NO. 5, 685-691.

7. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol., 1997, VOL.28, NO.3, pp 333-344.

8. Huen Y.K.: The twin prime problem revisited, INT. J. MATH. Educ. Sci. Technol., 1997, VOL.28, NO. 6, 825-834.

9. Huen Y.K.: Is Pi Periodic? INT.J. Math. Educ. Sci. Technol., 1998, VOL.29, No.1, 19- 26.

10. Huen Y.K: Final Value Theore In Number Sequences, INT.J. Math. Educ. Sci. Technol., 1998, VOL.???, No.???, ???-???. (in the press).

11. Huen Y.K.: Order Analysis In Sequence Algebra, INT.J. Math. Educ. Sci. Technol., 1998, VOL.???, No.???, ???-???. (in the press).

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