I have been writing over the past number of blogs regarding the nature of the Zeta 2 zeros which basically refer to the non-trivial roots (i.e. the roots other than 1) corresponding to prime numbered dimensions.
Once again these zeros initially arise in the context of finite solutions for
ζ2(s) = 1 + s1 + s2 + s3 +….. + st – 1 (with t prime) = 0.
So for example when t = 3, the relevant equation is
1 + s1 + s2 = 0, which has two unique solutions
i.e. s = – 1/2 + .866i and s – 1/2 – .866i
Then when extended in the (conventional) infinite manner,
ζ2(s) = 1 + s1 + s2 + s3 +….. = 0, provided we group terms in regular cycles of t.
If however we increase terms as is normally the case one at a time then the value for ζ2(s) = 1/2 for all (non-trivial) prime numbered root values of t.
And as we have seen this value of 1/2 expresses a probable value (in the same manner as the likelihood of Heads on the toss of an unbiased coin).
Now with respect to the Zeta 1 Function, according to the Riemann Hypothesis the non-trivial zeros are all postulated to lie on the imaginary line drawn through 1/2. However strictly this value expresses a probability (with an inevitable circularity in interpretation applying).
Ultimately such probabilities reflect an act of faith that actual finite behaviour is ultimately consistent with potential infinite behaviour.
So once again potentially for the general infinite case, the probability of Heads (on tossing an unbiased coin) = 1/2.
Now the relative frequency of the actual number of Heads in any finite number of trials is likely to deviate from this potential result.
Thus in effect the probability notion reflects the belief that ultimate consistency as between both sets of behaviour i.e. potential (qualitative) and actual (quantitative) will be maintained in a relative approximate manner.
However we clearly cannot prove this in the standard absolute fashion (based on merely reduced quantitative notions).
Now I wish again to emphasise the huge potential significance of these higher dimensions.
Firstly they open up an entirely new holistic appreciation of mathematical symbols (based directly on qualitative rather than quantitative recognition).
Secondly they provide the crucial means by which we are enabled to make ordinal distinctions as between numbers.
Again we must be clear from the outset of two distinct notions of number.
The Type 1 aspect relates to the cardinal interpretation of number in a collective quantitative manner (where individual members lack qualitative distinction).
Thus for example, 3 from a cardinal perspective reflects a collective number quantity (as a whole unit).
Though we can represent 3 in quantitative terms as 1 + 1 + 1, these individual units (as homogeneous) thereby lack any qualitative distinction.
The Type 2 aspect by contrast relates to the ordinal interpretation of number as unique qualitative units (where the collective sum of these units now lacks any quantitative distinction).
Now the Type 2 aspect is indirectly expressed in a quantitative manner through its corresponding roots of 1 (all of which - except 1 are uniquely defined for prime roots).
So the unique qualitative units of 3 are its 1st, 2nd and 3rd members (which arise through a mutual interdependence with each other).
Expressed in an indirect quantitative manner, these are 1, – 1/2 + .866i and – 1/2 – .866i
Then when the mutual interdependence of these is expressed through their sum, the quantitative value = 0.
Thus the Type 1 aspect is based on the quantitative notion of number as independent; the Type 2 aspect - by contrast - is based on the qualitative notion of number as interdependent.
However in dynamic interactive terms independence necessarily implies interdependence and interdependence, independence respectively.
Thus we cannot meaningfully place cardinal numbers quantitatively in relation to each other without an implied ordinal aspect; likewise we cannot meaningfully rank numbers ordinally without an implied cardinal aspect. So for example we must always fix the first member of a group in cardinal terms before other members can be meaningfully related to each other.
So the enormous significance of the Zeta 2 (non-trivial roots) is that they provide the true qualitative basis for the ordinal interpretation of number.
Thus implicit in our customary ordinal interpretation of number at the conscious level of quantitative interpretation is an unrecognised unconscious aspect that enables true recognition of number interdependence in holistic terms.
Thus the Zeta 2 zeros represent the (unrecognised) unconscious basis of the ordinal number system.
Properly understood the experience of number represents the dynamic interaction therefore of conscious (analytic) and unconscious (holistic) aspects.
So the natural numbers represent the conscious (analytic) aspect; the Zeta 2 zeros represent the unconscious (holistic) aspect with respect to its ordinal interpretation.
However because of our merely reduced conventional interpretation of the number system in quantitative terms, the vital role of the Zeta 2 zeros still remains completely unknown.
Friday, October 11, 2013
Wednesday, October 9, 2013
Where Science and Art Coincide (8)
So far we have confined ourselves with solutions with respect to the finite expression of the Zeta 2 Function,
ζ2(s) = 1 + s1 + s2 + s3 +….. + st – 1 (with t prime) = 0
Now this can be extended in an (conventional) infinite manner provided we only take groups of t terms.
So for example in the simplest case where t = 2 (where the finite solution is s = – 1) , the infinite expression,
ζ2(s) = 1 + s1 + s2 + s3 +….. = 0,
will still have the solution – 1, provided that we maintain an even number of terms by extending in pairs.
However the problem then arises as to what the value of the expression ζ2(s) will be when we remove this restriction that terms be taken in groups of t.
Again in the simplest case where s = – 1, we generate the alternating series,
ζ2(s) = 1 – 1 + 1 – 1 + 1 – 1 +......
There are only two possible values to this infinite series.
If we take an odd number of terms, ζ2(s) = 1;
however if we take an even number of terms ζ2(s) = 0.
Since the probability of an even or odd number of terms is equal, then we could postulate the probable value of the series as the mean of both values = 1/2.
Fascinatingly 1/(1 – s) = 1 + s1 + s2 + s3 +…..
and when we put in the value s = – 1 in the expression we get
1/2 = – 1 + 1 – 1 + 1 – 1 +......
So the value here obtained by the formula (though non-intuitive in linear terms) provides the same answer as we earlier achieved through probabilistic reasoning.
This would therefore strongly suggest that the value 1/2 is a numerical value representing a probability.
We can put it even more strongly by saying that this very notion of probability is inherent in the Riemann Hypothesis, so that our standard probabilistic notions only hold on the basis that the Riemann Hypothesis likewise holds.
When for example we say that the probability of getting"Heads" in tossing an unbiased coin = 1/2, there is a unavoidable element of circularity in this definition.
In fact the notion of probability here entails the assumption of a consistent relationship as between finite and infinite notions.
Clearly when we toss a finite number of coins the actual frequency of the number of Heads is likely to deviate from 1/2.
Then the assumption that the result will tend ever closer to 1/2 (as we keep increasing the number of tosses), in itself requires the initial assumption that "Heads" and "Tails" are equally likely!
Thus what really lies behind the notion of probability is the fundamental issue of the consistency of finite and infinite notions (which are qualitatively distinct).
So when we maintain that the probability of obtaining "Heads" = 1/2, we initially intuit this in infinite terms (as potentially applying in all cases).
Then we attempt to maintain consistency as between this infinite result (that potentially holds) and finite behaviour (that applies in actual cases). So underlying our conclusion, that in repeated finite trials using an unbiased coin, the result will tend ever closer to the potential infinite result ( = 1/2) is the implicit assumption that finite and infinite behaviour do indeed correspond with each other.
Thus the finite (actual) behaviour of repeated finite trials of an unbiased coin is ultimately assumed to be fully consistent with its assumed infinite behaviour (in potential terms).
And indeed this would make sense. As I have stated previously linear are based on absolute notions. However circular (dimensional) notions are bases on relative notions.
However this truth can only be maintained in a relative approximate manner.
Indeed one could validly maintain that the assumption of the Riemann Hypothesis, that all the trivial non zeros (of the Zeta 1 Function) lie on the line with real part = 1/2, is exactly the same assumption that allows us to maintain that the probability of getting "Heads" with an unbiased coin = 1/2.
However this proposition cannot be proven in any absolute fashion as the finite behaviour of the coins can only approximate this assumed infinite behaviour in a relative manner.
So proof in conventional mathematical terms assumes an absolute (reduced) correspondence as between actual finite and potential infinite behaviour. Thus the truth of a general proposition e.g. the Pythagorean Theorem as potentially applying in all (infinite) cases is assumed to directly correspond with all (finite) cases in actual terms!
However with the simplest case of probability i.e. the probability of "Heads" or "Tails" on tossing an unbiased coin, we can see that the correspondence as between finite and infinite behaviour is merely of a relative approximate nature.
Though mathematicians may be loath to admit this - as it undermines all their basic assumptions regarding the nature of mathematical proof - the behaviour of the primes with respect to the natural numbers is likewise of a merely relative approximate nature, entailing the interaction as between infinite (qualitative) and finite (quantitative) notions.
Now it still remains a wonderful mystery how consistency is still preserved as between finite (analytic) and infinite (holistic) notions (when their distinctive nature is properly appreciated).
However this consistency is now necessarily of a probable rather than absolute nature. And again the Riemann Hypothesis establishes precisely the condition (with respect to the behaviour of the primes) for this consistency. So the relevance of 1/2 in this context (just like our chance of "Heads") is strictly of a probabilistic nature.
Again in returning to our formula,
1/(1 – s) = 1 + s1 + s2 + s3 +….. ,
when the value given in the formula seems non-intuitive from a conventional linear perspective, this implies that we must now switch to The Type 2 notion of number (which represents a dynamic interaction, with a merely probable numerical value).
This is a very significant point for - as we shall see - the value here of 1/2 ultimately is the basis for the value of the real part of the Zeta 1 zeros = 1/2, on which the Riemann Hypothesis centres.
The importance of this can be demonstrated for the Zeta 1 Function in providing a way for calculating its (non-intuitive) values, where s < 1.
Indeed I have used it in another blog entry to manually calculate the values for ζ1( – 1), ζ1( – 2) and ζ1( –3) respectively.
However an even more surprising factor is that when any of the possible root values are substituted for s in the infinite Zeta 2 Function, the same answer of 1/2 is obtained.
This needs some explaining so I will illustrate now with reference to the next prime number 3 (for its two non-trivial roots).
These two roots are – 1/2 + .866i and – 1/2 –.866i respectively.
Thus there are 3 possible options in terms of the number of terms in the series i.e. t, t + 1 and t + 2 respectively.
Now if with respect to the 1st possible value i.e. – 1/2 + .866i, we take the first 3 terms the sum of series = 0.
So for example s = – 1/2 – .866i and s2 = – 1/2 + 866i.
Therefore the sum of 1st 3 terms = 1 – 1/2 – .866i – 1/2 + 866i = 0. And this will repeat with succeeding groupings of 3 successive terms.
Then when we take t + 1 terms the sum = 1.
And when we take t + 2 terms, the sum = 1 – 1/2 + 866i. = 1/2 + .866i
However we can also calculate 3 possible values with respect to the other root, i.e. – 1/2 –.866i .
Once again the sum of first 3 successive terms = 1 – 1/2 –.866i – 1/2 +.866i = 0.
The sum of t + 1 terms = 1.
Finally the sum of t + 2 terms = 1 – 1/2 – .866i = 1/2 – .866i .
Thus is we now take the sum of the 6 possible values (using the two non-trivial roots in question) the total sum = 0 + 1 + 1/2 + .866i + 0 + 1 + 1/2 – .866i = 3.
Therefore as 6 values are included here the average mean value = 3/6 = 1/2.
Therefore the value of 1/2 once again represent the expected value (average mean value) over all the possible combination of terms with respect to corresponding non-trivial roots of t (where t is prime).
In fact there is another fascinating link here with the Zeta 1 Function.
As we have seen in our illustration for t = 3, the number of possible values = 3 * 2, i.e. t * (t – 1).
and earlier for t = 2, the number of possible values = 2 * 1.
However with respect to the natural number terms t, the frequency of primes = log t
Therefore if we use prime number groupings, the number of possible values = t * (log t – 1)
Then because these root values relate to circular - rather than linear - numbers when we now attempt to convert these to linear form by dividing t by 2π,
we obtain t/2π(log t/2π - 1) = t/2π.log t/2π - t/2π (which is the formula for calculation of the frequency of non-trivial zeros for the Zeta 1 Function).
Thus the trivial zeros for the Zeta 1 Function can thereby be seen as a linear expression (in imaginary number format) of the circularised version of the corresponding non-trivial zeros with respect to the Zeta 2 Function! And remember I have frequently expressed that the very significance of the imaginary notion in holistic mathematical terms is that it provides a manner of expressing notions that are inherently circular (paradoxical) in a linear format!
Remarkably therefore the two sets of zeros represent distinctive ways of expressing what ultimately represents the same identity i.e. differing perspectives on the same ultimate reality.
And these perspectives entail the dynamic interaction of both the quantitative (cardinal) and qualitative (ordinal) aspects of the number system, which can initially be viewed from two distinctive perspectives, but which ultimately are identical in a manner where experience of number approaches a pure ineffable state.
ζ2(s) = 1 + s1 + s2 + s3 +….. + st – 1 (with t prime) = 0
Now this can be extended in an (conventional) infinite manner provided we only take groups of t terms.
So for example in the simplest case where t = 2 (where the finite solution is s = – 1) , the infinite expression,
ζ2(s) = 1 + s1 + s2 + s3 +….. = 0,
will still have the solution – 1, provided that we maintain an even number of terms by extending in pairs.
However the problem then arises as to what the value of the expression ζ2(s) will be when we remove this restriction that terms be taken in groups of t.
Again in the simplest case where s = – 1, we generate the alternating series,
ζ2(s) = 1 – 1 + 1 – 1 + 1 – 1 +......
There are only two possible values to this infinite series.
If we take an odd number of terms, ζ2(s) = 1;
however if we take an even number of terms ζ2(s) = 0.
Since the probability of an even or odd number of terms is equal, then we could postulate the probable value of the series as the mean of both values = 1/2.
Fascinatingly 1/(1 – s) = 1 + s1 + s2 + s3 +…..
and when we put in the value s = – 1 in the expression we get
1/2 = – 1 + 1 – 1 + 1 – 1 +......
So the value here obtained by the formula (though non-intuitive in linear terms) provides the same answer as we earlier achieved through probabilistic reasoning.
This would therefore strongly suggest that the value 1/2 is a numerical value representing a probability.
We can put it even more strongly by saying that this very notion of probability is inherent in the Riemann Hypothesis, so that our standard probabilistic notions only hold on the basis that the Riemann Hypothesis likewise holds.
When for example we say that the probability of getting"Heads" in tossing an unbiased coin = 1/2, there is a unavoidable element of circularity in this definition.
In fact the notion of probability here entails the assumption of a consistent relationship as between finite and infinite notions.
Clearly when we toss a finite number of coins the actual frequency of the number of Heads is likely to deviate from 1/2.
Then the assumption that the result will tend ever closer to 1/2 (as we keep increasing the number of tosses), in itself requires the initial assumption that "Heads" and "Tails" are equally likely!
Thus what really lies behind the notion of probability is the fundamental issue of the consistency of finite and infinite notions (which are qualitatively distinct).
So when we maintain that the probability of obtaining "Heads" = 1/2, we initially intuit this in infinite terms (as potentially applying in all cases).
Then we attempt to maintain consistency as between this infinite result (that potentially holds) and finite behaviour (that applies in actual cases). So underlying our conclusion, that in repeated finite trials using an unbiased coin, the result will tend ever closer to the potential infinite result ( = 1/2) is the implicit assumption that finite and infinite behaviour do indeed correspond with each other.
Thus the finite (actual) behaviour of repeated finite trials of an unbiased coin is ultimately assumed to be fully consistent with its assumed infinite behaviour (in potential terms).
And indeed this would make sense. As I have stated previously linear are based on absolute notions. However circular (dimensional) notions are bases on relative notions.
However this truth can only be maintained in a relative approximate manner.
Indeed one could validly maintain that the assumption of the Riemann Hypothesis, that all the trivial non zeros (of the Zeta 1 Function) lie on the line with real part = 1/2, is exactly the same assumption that allows us to maintain that the probability of getting "Heads" with an unbiased coin = 1/2.
However this proposition cannot be proven in any absolute fashion as the finite behaviour of the coins can only approximate this assumed infinite behaviour in a relative manner.
So proof in conventional mathematical terms assumes an absolute (reduced) correspondence as between actual finite and potential infinite behaviour. Thus the truth of a general proposition e.g. the Pythagorean Theorem as potentially applying in all (infinite) cases is assumed to directly correspond with all (finite) cases in actual terms!
However with the simplest case of probability i.e. the probability of "Heads" or "Tails" on tossing an unbiased coin, we can see that the correspondence as between finite and infinite behaviour is merely of a relative approximate nature.
Though mathematicians may be loath to admit this - as it undermines all their basic assumptions regarding the nature of mathematical proof - the behaviour of the primes with respect to the natural numbers is likewise of a merely relative approximate nature, entailing the interaction as between infinite (qualitative) and finite (quantitative) notions.
Now it still remains a wonderful mystery how consistency is still preserved as between finite (analytic) and infinite (holistic) notions (when their distinctive nature is properly appreciated).
However this consistency is now necessarily of a probable rather than absolute nature. And again the Riemann Hypothesis establishes precisely the condition (with respect to the behaviour of the primes) for this consistency. So the relevance of 1/2 in this context (just like our chance of "Heads") is strictly of a probabilistic nature.
Again in returning to our formula,
1/(1 – s) = 1 + s1 + s2 + s3 +….. ,
when the value given in the formula seems non-intuitive from a conventional linear perspective, this implies that we must now switch to The Type 2 notion of number (which represents a dynamic interaction, with a merely probable numerical value).
This is a very significant point for - as we shall see - the value here of 1/2 ultimately is the basis for the value of the real part of the Zeta 1 zeros = 1/2, on which the Riemann Hypothesis centres.
The importance of this can be demonstrated for the Zeta 1 Function in providing a way for calculating its (non-intuitive) values, where s < 1.
Indeed I have used it in another blog entry to manually calculate the values for ζ1( – 1), ζ1( – 2) and ζ1( –3) respectively.
However an even more surprising factor is that when any of the possible root values are substituted for s in the infinite Zeta 2 Function, the same answer of 1/2 is obtained.
This needs some explaining so I will illustrate now with reference to the next prime number 3 (for its two non-trivial roots).
These two roots are – 1/2 + .866i and – 1/2 –.866i respectively.
Thus there are 3 possible options in terms of the number of terms in the series i.e. t, t + 1 and t + 2 respectively.
Now if with respect to the 1st possible value i.e. – 1/2 + .866i, we take the first 3 terms the sum of series = 0.
So for example s = – 1/2 – .866i and s2 = – 1/2 + 866i.
Therefore the sum of 1st 3 terms = 1 – 1/2 – .866i – 1/2 + 866i = 0. And this will repeat with succeeding groupings of 3 successive terms.
Then when we take t + 1 terms the sum = 1.
And when we take t + 2 terms, the sum = 1 – 1/2 + 866i. = 1/2 + .866i
However we can also calculate 3 possible values with respect to the other root, i.e. – 1/2 –.866i .
Once again the sum of first 3 successive terms = 1 – 1/2 –.866i – 1/2 +.866i = 0.
The sum of t + 1 terms = 1.
Finally the sum of t + 2 terms = 1 – 1/2 – .866i = 1/2 – .866i .
Thus is we now take the sum of the 6 possible values (using the two non-trivial roots in question) the total sum = 0 + 1 + 1/2 + .866i + 0 + 1 + 1/2 – .866i = 3.
Therefore as 6 values are included here the average mean value = 3/6 = 1/2.
Therefore the value of 1/2 once again represent the expected value (average mean value) over all the possible combination of terms with respect to corresponding non-trivial roots of t (where t is prime).
In fact there is another fascinating link here with the Zeta 1 Function.
As we have seen in our illustration for t = 3, the number of possible values = 3 * 2, i.e. t * (t – 1).
and earlier for t = 2, the number of possible values = 2 * 1.
However with respect to the natural number terms t, the frequency of primes = log t
Therefore if we use prime number groupings, the number of possible values = t * (log t – 1)
Then because these root values relate to circular - rather than linear - numbers when we now attempt to convert these to linear form by dividing t by 2π,
we obtain t/2π(log t/2π - 1) = t/2π.log t/2π - t/2π (which is the formula for calculation of the frequency of non-trivial zeros for the Zeta 1 Function).
Thus the trivial zeros for the Zeta 1 Function can thereby be seen as a linear expression (in imaginary number format) of the circularised version of the corresponding non-trivial zeros with respect to the Zeta 2 Function! And remember I have frequently expressed that the very significance of the imaginary notion in holistic mathematical terms is that it provides a manner of expressing notions that are inherently circular (paradoxical) in a linear format!
Remarkably therefore the two sets of zeros represent distinctive ways of expressing what ultimately represents the same identity i.e. differing perspectives on the same ultimate reality.
And these perspectives entail the dynamic interaction of both the quantitative (cardinal) and qualitative (ordinal) aspects of the number system, which can initially be viewed from two distinctive perspectives, but which ultimately are identical in a manner where experience of number approaches a pure ineffable state.
Monday, October 7, 2013
Where Science and Art Coincide (7)
Another way of looking at the dimensional number system is as a spectrum entailing the dynamic interaction of both quantitative and qualitative aspects (in relative terms).
From this perspective the only number that remains undefined on this spectrum is 1, as it is the only number where both quantitative and qualitative characteristics are absolutely similar.
Again, as the 1st root of 1 is 1, this is the only number where the corresponding roots of 1 are identical with the original dimensional number (i.e. power) of the number!
This would further imply that the correct dynamic interpretation of the Riemann Zeta Functions is that it in fact represents the number system (considered as a spectrum) with the only number undefined in this system where s (the dimensional number) = 1.
This (Type 2) dimensional number system also has intimate connections with Euler's Identity.
In fact Euler's Identity - or at least the slightly modified form I employ - provides the ready means of calculating all the various root values of 1 (which in this new interpretation have both quantitative and qualitative interpretations which are complementary).
Now Euler's Identity can be expressed as,
However the more fundamental version is obtained from squaring both sides so that,
e2iπ = 1
More correctly this can be expressed as
e2iπ = 11 I have explained the profound holistic importance of these number symbols before. e is unique in the sense that both its differential and integral are the same. Therefore in experiential terms e plays the role of the symbol where both (analytic) differentiation and (holistic) integration are ultimately identical.
Therefore for example 2 in the Type 1 system is expressed as 2e2iπ Then k in the Type 2 system is expressed as e(2iπ)k Therefore 2 in the Type 2 system is expressed as e(2iπ)2 = 2 i.e. 12 Now of course roots will be expressed as the reciprocals of whole numbers Now De Moivre's formula can be easily obtained from Euler's Identity. Therefore to obtain the corresponding roots of 1 (using the fundamental Euler Identity)
i.e. 1 + s1 + s2 = 0
From this perspective the only number that remains undefined on this spectrum is 1, as it is the only number where both quantitative and qualitative characteristics are absolutely similar.
Again, as the 1st root of 1 is 1, this is the only number where the corresponding roots of 1 are identical with the original dimensional number (i.e. power) of the number!
This would further imply that the correct dynamic interpretation of the Riemann Zeta Functions is that it in fact represents the number system (considered as a spectrum) with the only number undefined in this system where s (the dimensional number) = 1.
This (Type 2) dimensional number system also has intimate connections with Euler's Identity.
In fact Euler's Identity - or at least the slightly modified form I employ - provides the ready means of calculating all the various root values of 1 (which in this new interpretation have both quantitative and qualitative interpretations which are complementary).
Now Euler's Identity can be expressed as,
eiπ = – 1
Now with respect to the unit circle,
2π represents the circumference. However in analytic terms this will still have positive extension.
However if we try and envisage a circle whose radius simultaneously has the same positive and negative unit value, in dynamic terms this will shrink to a point where the (linear) diameter and (circular) circumference approach identity.
Therefore if we are to properly interpret the Euler Identity we must combine both Type 1 (analytic) and Type 2 (holistic) interpretations.
In spiritual contemplative terms the very means of achieving ultimate unity (which of course can only be approximated in dynamic terms) entails seeking this ultimate identity with respect to both finite (analytic) and infinite (holistic) understanding.
And as number - properly understood - represents the very basis of such experience, the very notion of 1 comprehensively understanding 1 (as numerical unity) entails the same process.
Now the very reason why this is not apparent in customary terms is precisely because Conventional Mathematics employs a merely reduced (i.e. absolute) analytic notion of number.
However the journey to understanding this notion of 1 in a comprehensive dynamic manner (where both analytic and holistic appreciation interact) is inseparable from the spiritual contemplative process of achieving unity of all experience.
So again properly understood the fundamental Euler Identity opens up a remarkable window of appreciation to both the Type 1 and Type 2 aspects of the number system.
So for example any number (k) in the Type 1 system can be expressed as
ke2iπ
e2iπ = cos(2π)
Therefore to get the k roots of 1 we obtain,
e2iπ/k = cos(2πk)
So for example to obtain the 3 roots of 1, we let k = 1/3, 2/3 and 3/3 respectively.
Therefore the 1st root for k = 1/3 is
e2iπ/3 = cos(2π/3)
= – 1/2 + .866 i
The 2nd root for k = 2/3 is
e4iπ/3 = cos(4π/3) + i sin(4π/2) = cos 240 + i sin 240
= – 1/2 – .866 i
The 3rd root for k = 3/3 is
e6iπ/3 = cos(6π/3) + i sin(6π/2) = cos 360 + i sin 360
= 1
There is an intimate connection here with the (finite) Zeta 2 Function
The Zeta 2 Function represents all the non-trivial solutions for roots i.e. all those except 1)
Therefore the first 2 solutions above for k = 1/3 and 2/3 represent the solutions for the Zeta 2 Function (where t = 3)
ζ2(s) = 1 + s1 + s2 + s3 +….. + st – 1 (with t prime) = 0, where t = 3
i.e. 1 + s1 + s2 = 0
Subscribe to:
Posts (Atom)