The Two Alternative Versions Of Zeta Function - are these exactly equivalent?

The Two Alternative Versions Of Zeta Function - are these exactly equivalent?

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: 8/7/98)

Abstract
Two alternative versions of zeta function were defined by Euler, one of which is the summative version whilst the other is the product version. In the s-domain, the two versions look so different that it is difficult to validate whether these are exactly equivalent. By mapping these two formulations into the z-domain, it is possible to compare the similarities or otherwise between these two sequences after taylor series expansions. Theoretically this should settle the equivalence or otherwise between these two versions, but not quite. The reason is that mapping from the s-domain to the z-domain is not 1-1 and onto since in the latter, zeros are suppressed. Therefore we can only compare the two sequences with the zeros excluded. The conclusion is that the method of determining equivalence between two sequences in the z-domain is incomplete but it is still a big improvement over comparing them in the s-domain.

1. Introduction

Riemann's Hypothesis remains one of the most important unsolved problem in this centuary. From a sequence algebraic point of view, the difficulties in solving this problem could be attributed to the awkward format defined for the summative version of zeta function as shown in equation (1). It is difficult to find a closed formulation for a sequence with terms such as k^s in the s-domain. This form also makes it difficult to find the roots analytically. Hence we can only evaluate zeta(s) numerically by supplying values to s. To prove that the two functions are identical, one must derive analytical expressions for predicting the roots of zeta and when both sets are identical then the two functions must be equivalent. This seems reasonable but is difficult to realise. If two closed sequence algebraic functions are series expanded and these generate the same sequence, then the two functions must be identical. Again one must have at hand closed sequence algebraic forms ready for taylor series expansions. In the present paper, it is found that the second option is simpler because both versions can be mapped into the z-domain and from it closed formulations can be found.

..ub
.-----
..\.......1................1.........1........1........1.........1........1........1........1........1
...)....------ = 1 + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + --- ........(1).
../...........s.................s.........s........s.........s.........s.........s.........s.........s........s
.-----..k................2.........3........4........5.........6........7........8........9.......10
..k = 1

If the above sequence is transformed into the z-domain, then the prospect of finding a closed form as shown in equation (2) is much brighter although now terms are plotted along a logarithmic axis with a suppressed zero. Nevertheless so long as one is aware of this pitfall, much insights can be obtained from the mappings. To transform equation (1) into equation (2) in the z-domain, one makes use of the relation z = E^T*s with T assigned as unity, so that z = E^s. Assigning T = 1 is a standard number theoretic procedure in sequence algebra but in the seiries expansions of zeta function, the intervals between successive terms are irrational and which must be approximated by rational fractions.

..ub
.-----
..\.......1................1........1........1........1.........1.......1........1........1.........1
...)....------ = 1 + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + --- ........(2).
../.........ln(k)............ln(2)...ln(3)...ln(4)...ln(5)...ln(6)...ln(7)...ln(8)...ln(9)....ln(10)
.-----..z................z.........z........z.........z.........z........z........z.........z.........z
..k = 1

The zeta function, when mapped into the z-domain, shows that it can be interpreted in sequence algebra as representing the integer number set ( 1,2,3,....... } plotted along a logarithmic axis with the zero origin suppressed [1 to 7]. This is because 1/z^ln(1) = 1 whereas 1/z^ln(0) is indeterminate. The main difficulty in attempting to find a closed form from the z-domain is that the mapping is almost but not quite 1-1 and onto. Comparison can only be made with the understanding that the zeros are excluded in sequences to be compared.

The closed sequence algebraic formulation for the summative version of zeta function in the z-domain is given by equation (3).

........ub
.......-----
........\..............1
1 + ...)....------------- ........(3).
......../...........ln(k).........
.......-----...z........ - 1
......k = 1

Taylor series expansions of the open form in equation (2) is given by equation (4)

eval(1 + sum(1/z^(s*ln(k)),k,1,6));

...1..................1...................1...................1....................1
------------ + ------------ + ------------ + ------------ + ------------+ 1.0 ........(4).
..... 0.69315 ..... 1.09861 ..... 1.38629 ...... 1.60944 ....... 1.79176
....z..................z....................z....................z....................z

Taylor series expansions of the closed form in equation (3) is given by equation (5)

float(eval(1 +taylor(sum(1/(z^(ln(k))-1),k,2,7),z,inf,7)));

....1...............1................2.0..............1.................1
---------- + ---------- + ---------- + ---------- + ---------- + 1.0 + . . . . ....(5).
....0.69315.... 1.09861.... 1.38629...... 1.60944..... 1.79176
...z................z.................z..................z.................z

Comparisons of the two sequences in equations (4) and (5) show that these are not exactly identical, the differences being in the numerator coefficients. There is an explanation for this arising from product of sequences. When two sequences are multiplied, we are actually working in a 2-dimensional space. However, in number theoretic problems, we always map this back into a 1-dimensional space. Diagonal terms are unique but off-diagonal terms always come in image pairs. This explains why some numerator coefficients are greater than unity. In sequence algebra, we introduce a normalising function called Normc to reduce all numerator coefficients back to unities. This is necessary because we are only interested in order properties along a 1-dimensional axis. After normalising, then the two sequences are found to be identical as shown in equation (6).

Normc(float(eval(1 +taylor(sum(1/(z^(ln(k))-1),k,2,7),z,inf,7))));

....1...............1................1.................1.................1
---------- + ---------- + ---------- + ---------- + ---------- + 1.0 + . . . . ....(6).
....0.69315.... 1.09861.... 1.38629...... 1.60944..... 1.79176
...z................z.................z..................z.................z

The section shows that it is possible to map open formulations from the s-domain into the z-domain and yet be able to derive the closed form in the latter. The procedure is to be applied to compare the equivalence between the summative zeta form and the product zeta form in the next section.

2. Equivalence Of Summative And Product Form

One interesting property connected with zeta function is that there is an alternative product formulation involving primes given by equation (7).

This formulation can be mapped into the z-domain using the relation p^s = z^ln(p) as shown in equation (8).

If equation (8) is compared with equation (4), one would recognise some similarities of algebraic form with the exception that in the former, only primes are used in the expansions.

To expand equation (8) first convert it into a form involving only primes as shown in equation (9).

zeta_prod:product((z^ln(k)/(z^ln(k)-1))^((primep(k)-false)/(true-false)),k,2,7);

.........................................5.34711
.......................................z
---------------------------------------------------------------....................(9).
...0.69315..........1.09861...........1.60944...........1.94591
(z............. - 1) (z.............. - 1) (z.............. - 1) (z.............. - 1)

Taylor series expansions using Macsyma 2.2.1 will give a sequence in equation (10) of which only the first six terms are to be compared with those from equation (4). It should be pointed out that not all symbolic packages will accept decimal indices for taylor expansions. For example, Maple V R 3 will reject the above formulation.

zeta_prodseq:eval(float(taylor(zeta_prod,z,inf,7)));

RAT replaced 5.34711 by 647/121 = 5.34711
RAT replaced 1.94591 by 1475/758 = 1.94591
RAT replaced 1.60944 by 5184/3221 = 1.60944
RAT replaced 1.09861 by 3721/3387 = 1.09861
RAT replaced 0.69315 by 1143/1649 = 0.69315
RAT replaced 0.69315 by 1143/1649 = 0.69315
RAT replaced 1.09861 by 3721/3387 = 1.09861
RAT replaced 1.60944 by 5184/3221 = 1.60944
RAT replaced 1.94591 by 1475/758 = 1.94591

The difference between the two sequences is minor in equation (10) where the unity term is approximated by 1/(z^(6.18278e-7))) due to numerical approximations introduced by the symbolic pacakge.

(1/(z^(6.18278e-7))) + (1/(z^0.69315)) + (1/(z^1.09861))
+ (1/(z^1.38629)) + (1/(z^1.60944)) + (1/(z^1.79176))
+ (1/(z^1.94591)) + (1/(z^2.07944)) + (1/(z^2.19722))
+................................................................+...........
+ (1/(z^6.87936)) + (1/(z^6.88755)) + (1/(z^6.90775))
+ (1/(z^6.91573)) + (1/(z^6.93147)) + (1/(z^6.93634))
+ (1/(z^6.95655)) + (1/(z^6.98472))..................(10).

Thus the summative version and the product version of zeta functions are in agreement if we ignore the suppression of zeros and concentrate on the order properties by ignoring duplicities in the numerator coefficients in series expansions of the latter.

3. Conclusions

This investigation shows that it is possible to map open forms in the s-domain to the z-domain from which closed formulations can be developed. This is a handy procedure when it is found difficult to check the equivalence between two algebraic formulations in the s-domain. In the present case, the summative form of zeta function is algebraically awkward to find a closed form. In mappings, one must accept compromise in that this is not 100% 1-1 and onto since the zeros in the s-domain are suppressed in the z-domain due to the use of logarithmic scales. One should also recongise that in sequence algebra, two properties, viz., duplicity and order appear simultaneously. One needs only compare order properties by ignoring duplicities of numerator coefficients. The investigations show that the summative form and the product form of zeta function are order equivalent.

4. References

1. 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.

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

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

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

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

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

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

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