Tuesday, January 31, 2012

Connecting Positive (Even) and Negative (Odd) Integers of Zeta Function

I have discovered for myself a very interesting relationship which - based on initial empirical evidence - I would conjecture, universally holds.

The first part can be stated in the following manner.

If s (representing the dimensional power) of the Riemann Zeta Function is a positive even integer and if the denominator, i.e. D1 of ζ(1 - s) is divisible by s + 1 then, s + 1 is prime.

Furthermore in this case the denominator of ζ(s) i.e. D2 will be divisible also by s + 1.

Then if ai (from i = 1 to k) represent all the factors of D1, D1 will be divisible by ai + 1 (from 1 to k) whenever ai + 1 is a prime.
Furthermore in such cases D1 will have no other prime factors (though multiples of some existing factors may occur).

Also in many cases when D1 is divided singly by all of the prime factors (ai + 1), frequently the result = s.
Also the probability that s will result seems to be related to the degree to which s is of a very composite nature (i.e. with many factors). I carried out calculations for all negative odd integers up to 100 to find that half of the results (after division of denominator by the relevant prime factors) = s!

Likewise D2 will be divisible by all prime numbers from 3 to s (and only these primes) though in the case where s represents a power of 2, D2 will be divisible by all primes from 2 to s + 1 (and only these primes).

Even when D2 is not divisible by s + 1, or in the cases where s is an odd integer (> 1), D2 will be divisible by all prime numbers from 2 to the largest prime immediately prior to s + 1!


So to illustrate!

s = 10 is an even integer.

Therefore if the denominator of ζ(- 9) is divisible by 11, then 11 is prime.

The relevant denominator = 132 (in absolute terms) and 132 is indeed divisible by 11.

Therefore 11 is prime.

This also means that the denominator of ζ(10) will be divisible by 11.

And D2 = 93555 which again is divisible by 11.


The factors of 10 are 1, 2, 5 and 10.
When we add 1 to each factor we get 2, 3, 6 and 11.

So the prime numbers here are 2, 3 and 11.

Therefore 132 is divisible by 2, 3 and 11 (and only these primes).

The remainder that results on division by these prime numbers = 2 (≠ s).

So 132 is not an especially composite number (with 3 prime factors).

Also 93555 is thereby divisible by 3, 5, 7 and 11 (and only these primes).


So hidden in the structure of the rational denominator values of the Riemann Zeta Function, for negative odd integer values, is a simple test to determine whether a number is prime! Though of course it does not represent a practical way of finding prime numbers, it remains however an extremely interesting fact!

And it is no less interesting that for positive even integer values of s (regardless of how large is the value of s) that when s + 1 is prime, the denominator of ζ(s) will be divisible by all primes from 3 to s + 1 (and only those primes). Again where s is a square of 2 the denominator of ζ(s) is divisible by all primes from 2 to s + 1 (and only these primes).


addendum (added 29/4/2012)

It probably would help to illustrate with refernce to a couple of more examples.

s = 30 is an even integer.

Therefore if the denominator of ζ( - 29) is divisible by 31, then 31 is prime.

The relevant denominator = 85932 (in absolute terms) and 85932 is indeed divisible by 31.

Therefore 31 is prime.

This also means that the denominator of ζ(30) will be divisible by 31.

And D2 = 5660878804669082674070015625 is indeed divisible by 31.


The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.
When we add 1 to each factor we get 2, 3, 4, 6, 7, 11, 16 and 31.

So the prime numbers here are 2, 3, 7, 11 and 31.

Therefore 85932 is divisible by 2, 3, 7, 11 and 31 (and only these primes).



The remainder that results on division by these prime numbers = 6 (≠ s).

Perhaps it is a liitle surprising here that s = 30 is not the remainder, as 85932 is quite a composite number (though with no prime factors from 13 to 29 inclusive)!


The denominator of D2 (5660878804669082674070015625) will however in this case be divisible by all prime factors from 3 to 31 (inclusive) and only these factors.


In fact 5660878804669082674070015625 = 3^15.5^6.7^5.11^3.13^2.17.19.23.29.31

When however s = 32 the denominator of ζ-(31) is not now divisible by s + 1 = 33

i.e. the absolute value of denominator of ζ-(31) = 16320 (which is not divisible by 33). So 33 therefore is not prime!

In fact not only can we directly confirm whether a number is prime (through division of the relevant denominator) of the Zeta Function (as described above) we can also confirm directly that a number is not prime (as illustrated in this case whenever the denominator is not divisible by the number in question).


However because 32 is a square of 2, this implies that the denominator of ζ(32) will in this case be divisible by every prime number from 2 to 31 inclusive and only these prime factors (though of course these prime factors may repeat).

And this denominator

= 64290220571022341207266406250 = 2.3^15.5^8.7^4.11^2.13^2.17^2.19.23.29.31

Monday, January 30, 2012

Further Thoughts on Non-Trivial Zeros

It struck me forcibly over the weekend that the relationship between the distribution of prime numbers and the corresponding distribution of non-trivial zeros is even more intimate than I had realised.

Therefore it is very easy to move from a knowledge of the distribution with respect to one distribution to corresponding knowledge with respect to the other.

For example the average spread of prime number in the region of n is given simply as log n.

Therefore for example in the region of hundred the average spread (or gap) as between prime numbers is 4.605 (approx).

Now if we let n = t/2pi, we can correspondingly obtain the average gap as between the trivial non-zeros in the region of t = n*2pi = 628.3(approx).

This gap is given by the formula 2pi/{log (t/2pi)} = 6.283/4.605 = 1.3644 (approx).

So we can say therefore that the average spread as between non-trivial zeros in the region of 628 = 1.364 (approx).

So to obtain this latter answer (in the region of t) we simply divided 2pi by the average spread as between primes (in the region of n)!


We could of course equally calculate the average spread as between primes (for any value of n) from the spread as between non-trivial zeros (for the corresponding value of t).

As we have seen we can calculate the change in the average spread as between primes as n increases to n + 1 by the simple expression 1/n.

Therefore by we would expect the average spread as between primes to increase by 1/100 as we move from 100 to 101 (more correctly from 99.5 to 100.5).

A corresponding expression can be obtained through differentiation with respect to t in the original expression for calculating the gap as between non-trivial zeros,

= -(4*pi^2)/{log [(t/2pi)^2]*t}. This is negative as the average gap between non-trivial zeros decreases as the value of t increases!

So once again where n = 100 = t/2pi, it works out as -.00296 (approx)

Therefore in moving from t to t + 1, the average gap as between non-trivial zeros declines by .003 (approx).

Sunday, January 22, 2012

The mystery of e

For some 40 years or so I have been attempting to develop to my own satisfaction a distinctive type of qualitative Mathematics i.e. Holistic Mathematics. Recently I have referred to this greatly neglected aspect as Type 2 Mathematics. So just as Conventional (Type 1) Mathematics is devoted to the specialised development of the quantitative, equally in a balanced approach Holistic (Type 2) Mathematics would be correspondingly devoted to the specialised development of the (neglected) qualitative aspect.

Then finally, we would have the Comprehensive (Type 3) Mathematics where both quantitative and qualitative aspects would be dynamically combined in a highly productive and creative manner.

Though initially I spent many years developing holistic mathematical concepts in considerable isolation from conventional mathematical pursuits, recently I have begun to see in a much clearer manner, how a merely reduced interpretation is given in Type 1 Mathematics of fundamental relationships.

Ultimately both Type 1 and Type 2 aspects - because of their mutual interdependence - can only be fully appreciated in the context of comprehensive (Type 3) understanding.


For example I have pointed out elsewhere how the notion of the infinite cannot be successfully incorporated in standard (linear) rational terms; also I have pointed to the fact that when we raise a number to a (dimensional) power or exponent, that the number representing the exponent is properly speaking of a qualitative nature (in relation to its base number quantity). So this begs the obvious question of how even a simple procedure such as the squaring of a number (which entails both quantitative and qualitative aspects) can be properly interpreted in Conventional (Type 1) terms!


Yet another important example of the same problem is provided by irrational numbers. The very point regarding an irrational number such as the square root of 2 is that it combines both finite and infinite aspects of behaviour. So in quantitative terms the number can be approximated in discrete rational terms to any required degree of accuracy. However in qualitative terms its exact (quantitative) value can never be known with a decimal sequence that continues indefinitely in no fixed order.

In fact there is a very deep reason why the Pythagoreans found the discovery of the the existence of the square root of 2 so devastating. They could see - much more clearly than in modern times - that the nature of this number literally transcended their (Type 1) rational paradigm of Mathematics. In other words there was no way using this paradigm that the qualitative aspect of an irrational number could be successfully accommodated.

In subsequent Western Mathematical history, this issue has not been solved but rather avoided through a continual specialisation in the process of quantitative type reductionism.


The well known number e is an irrational (transcendental) number that combines both quantitative and qualitative aspects in a very special manner. And it is the appreciation of the nature of this quantitative/ qualitative type identity of e that is key to understanding the true nature of prime numbers.


Now it is well known in simple calculus that if y = e^x, that dy/dx = e^x.


Now differentiation and integration also have a qualitative holistic significance as is apparent in psychological development. So basically in experience, the differentiation of meaning pertains to the (rational) conscious, whereas integration pertains to the (intuitive) unconscious. In other words differentiation is discrete and quantitative in nature; by contrast integration is continuous and qualitative in nature.

Before development commences, in a sense differentiation and integration are identical (in total confusion) as mere potential for existence. Then - as is portrayed well in the mystical literature - again differentiation and integration once again approach identity with each other in pure spiritual union (where discrete phenomena are now so short-lived and refined that they no longer appear to even arise in experience!


What is truly remarkable is the intimate connection which e has with the prime numbers.

So for example the probability that a number is prime is given as 1/log n.
Alternative the average gap (or spread) between primes in the region of n is given as log n.

Now if e^x = n then log n = x.

So if we allow x to take on the value of any positive real number > 1 then x is telling us the average gap between primes in the region of n (i.e. e^x). And the accuracy of this estimate improves steadily for larger n! (Of course because prime numbers are positive integers we would need to round the value of n to the nearest integer!)


Now we have seen that because of the invariant characteristics of e^x relating to differentiation and integration both with respect to (conventional) Type 1 and (holistic) Type 2 interpretations, that e uniquely combines quantitative and qualitative characteristics in its own identity.

Likewise the fundamental nature of prime numbers is that they too combine both quantitative and qualitative characteristics in a unique manner. So this is the precise reason why e has such an intimate relationship with the primes!

So the Riemann Hypothesis - when correctly interpreted - represents a fundamental condition for maintaining the consistent identity of both the quantitative and qualitative aspects of the primes!

And this identity clearly points to the ineffable state in physical terms before creation begins where the prime number code still potentially awaits unveiling. Once we begin to differentiate distinct prime numbers then it is no longer possible to maintain total identity of discrete primes with the corresponding continuous general distribution of primes. So in a sense it is only before differentiation takes place in a phenomenal manner that an exact identity can be maintained of the discrete primes with their general distribution in a potential manner. It also points to the corresponding ineffable state spiritually at the peak of mystical union where the full mystery of the nature of the primes can finally be actualised.

Saturday, January 21, 2012

Music of the Primes

As we know Marcus du Sautoy gave the title "The Music of the Primes" to his book on the Riemann Hypothesis.

Interestingly, I see that Keith Devlin gives the same title to his Chapter on "The Riemann Hypothesis" in his book "The Millennium Problems".

It started with the Pythagoreans that discovered that the terms in the series 1,. 1/2, 1/3, 1/4, .. have a striking relevance for musical sound.
So this series was called the harmonic series.

This very series then provided the base for what ultimately was to become the Riemann Zeta Function which was shown to have a special relevance for the distribution of the primes. And then Riemann showed how to magically correct the general distribution of the primes with the wavelengths associated with the famed non-trivial zeros of the Zeta Function so as ultimately predict the exact number of primes up to a given number.

So one again there is a close relationship as between these wavelengths and musical harmonics.

So it is not perhaps surprising therefore that the term "the music of the primes" has proven so popular.


However the point I would make is that whereas prime numbers are conventionally interpreted in a quantitative manner, clearly the musical connotation with respect to the overall behavior of the primes has a qualitative (rather than quantitative) significance.

So therefore, properly understood the primes have both quantitative and qualitative aspects (which are interdependent).

Thus the key issue therefore with respect to the primes relates to the ultimate reconciliation of the quantitative with the qualitative aspect.

Once more as Conventional (Type 1) Mathematics relates merely to the quantitative aspect, it cannot satisfactorily hope to explore the true mystery of the primes.


Indeed as so often stated on this blog, the Riemann Hypothesis relates to the key condition necessary for the ultimate identification of both the quantitative and qualitative aspects of the primes. As the qualitative aspect is formally ignored by the mathematical profession, the Riemann Hypothesis cannot be proved (or disproved) in conventional terms. Indeed it points to an ineffable physical state which can only be fully understood in a corresponding ineffable spiritual manner.

And in this state the true mystery of the primes ultimately resides!

Thursday, January 19, 2012

Further Considerations

I thought it would be interesting to calculate the value of t where the average spread as between prime numbers and non-trivial zeros would be identical.

This would be given by the equation,

log t = 2π/log(t/2π) (It has to be remembered that whereas t on the LHS of equation corresponds to a real scale (representing quantities), t on the RHS corresponds to an imaginary scale (representing dimensional values).

After a bit of iteration, I found that the value for t corresponds to 36.2 (approx). Here the average gap as between prime numbers = 3.56 (approx); equally the average gap as between non-trivial zeros is also = 3.56 (approx).

It is fascinating once more to recognise that we can provide simple rules for switching from the average spread for one distribution (e.g. as between primes) to the corresponding average spread for the other distribution (i.e. as between non-trivial zeros).

So if we start with the average spread as between primes (in the region of t) i.e. log t, firstly we invert it to obtain 1/log t. Now we rewrite this as 1/(log t/1).

Then we replace the numerator 1 with 2π and the corresponding 1 (as denominator of log t also by 2π) So now we have 2π/(log t/2π) which gives us the estimate of the average spread as between non-trivial zeros (also in the region of t).

This would imply that the two approaches are in a sense mirrors of each other. So we can use the distribution of the (real) primes to infer the corresponding distribution of the (imaginary) non-trivial zeros. Equally in reverse fashion we can use the distribution of the (imaginary) non-trivial zeros to infer the corresponding distribution for (real) primes.


One further insight strikes me here. As is well known the non-trivial zeros can be effectively used to eliminate deviations (from actual) of the predicted general distribution of (real) prime numbers.

Therefore in principle it should be possible in reverse fashion to use the (real) primes to likewise eliminate the deviations (from actual results) of the predicted general distribution of the (imaginary) non-trivial zeros.


Finally, it is quite clear from the above that both the primes and the non-trivial zeros are in fact interdependent with each other representing a real to imaginary (and imaginary to real) relationship.

Now this might be easy enough to appreciate in quantitative terms, but of course in a balanced interpretation we must also include the qualitative counterparts of these very notions.

Put another way, it implies that quantitative nature of the primes (and their relationship to the natural numbers) is ultimately indistinguishable from corresponding qualitative interpretation of the primes (and their relationship to the natural numbers).

So once again the Riemann Hypothesis serves as the necessary condition for the reconciliation of both types of meaning. And this perfect reconciliation can only take place in an ineffable manner (where quantitative and qualitative are no longer distinguishable).

So the quest to know the ultimate nature of the prime numbers (in quantitative terms) - rightly understood - is inseparable from the corresponding quest to know likewise its nature in qualitative terms (both aspects which are interdependent)

So true knowledge of this ultimate mysterious nature (quantitative and qualitative) takes place in an ineffable manner (through perfect contemplative spiritual realisation).

Tuesday, January 17, 2012

Brief Further Addendum

It further struck me after completing the last entry that we could apply the same condition to the distribution of the primes as the distribution of the non-trivial zeros.

For example we could pose the question: When does the frequency of occurrence of prime numbers change by just 1 which equally is the condition that the probability of a number being prime = 1.

Now the probability of a number being prime can be approximated as 1/log t. So for this to equal 1 then t = e, i.e. 1/log e = 1/1 = 1.

So once again here we can see how e is absolutely central to prime distribution.

In fact we could express 1/log e in an alternative manner which is more revealing as to its true nature.

1 = e^0 whereas e = e^1

Therefore 1/log e = e^0/log (e^1).

So the probability that a prime number = 1, really relates to the fact that the nature of a prime equally combines linear (1) and circular (0) aspects of understanding which are perfectly enshrined in the notion of e.

So e^0 = 1; log e^1 = 1.

Addendum on Frequency of Non-Trivial Zeros

In the last post I mentioned the formula (which is surprisingly accurate) for the calculation of the number of non-trivial zeros up to a certain number.

This formula is

N(t) = t/2π*log(t/2π) - t/2π + O(log t)

Now again ignoring the final error term we can use this to calculate that the number of non-trivial zeros - for example - up to t = 100, is 28.127 (as against the correct prediction of 29).

As the calculation of the change in the spread as between primes in the region n (i.e. as n increases by 1) is given by the remarkably simple estimate of 1/n, it would therefore be very interesting to find a similar type estimate in the change of the spread as between the non-trivial zeros in the region of t.

In other words as t increases by 1 to t + 1, what is the change in the frequency of occurrence of non-trivial zeros?

Using my somewhat rusty application of rules in differentiating the formula at the top with respect to t, I came up with the estimate 1/2π(log t/2π).

In other words as t increases to t + 1, the change in the frequency of occurrence of the non-trivial zeros is given as 1/2π(log t/2π).


And this formula is simply the inverse of that which we have devised for calculation of the average spread in the frequency of non-trivial zeros.

So in fact, as we already calculated if the average gap as between trivial zeros in the region of 100 = 2.27, then the change in the frequency of non-trivial zeros per unit change in t is given as the inverse 1/2.27 = .44 (approx).


In particular this would imply that if we find the value of t for which the average spread as between non-trivial zeros = 1, then likewise the change in frequency of such zeros (in increasing from t to t + 1) will also be equal to 1.

Now earlier in the last post we saw that the formula for calculation of the average gap or spread as between non-trivial zeros is given as 2π/(log t/2π).

So for this value of t, 2π/(log t/2π) = 1
therefore 2π = log (t/2π)

so e^2π = t/2π {or remarkably 1^(-i) = t/2π}


So t works out at 3364.6 (approx) = 2π{1^(-i)} or 2π/(1^i)

So when t = 3365 (rounded to nearest integer) both the average gap in t for each new non-trivial zero and the change in the total frequency of non-trivial zeros (as t increases by 1) is approximately identical.


And finally, one must comment again on the similarity as between formulae with respect to behaviour of the primes on one hand and the behaviour of non-trivial zeros on the other. In each case the resemblance can be clearly shown through a linear to circular (or circular to linear) transformation on the one hand and then a simple inversion.

So the formula we worked out to calculate the change in frequency of the non-trivial zeros (as t increases to t + 1) is given as 1/2π(log t/2π).

Once again when we substitute the circular circumference of the unit circle 2π with its linear radius we obtain log t.

Now log t measures the spread as between primes in the region of t (where t here is measured on the real number scale), whereas its differential 1/t measures the change in this spread (as t increases to t + 1).


And once again I personally find it truly astonishing that the value for t (where the average gap between non-trivial zeros = 1 (which equally is the value where the frequency on such zeros increases by 1 as t goes to t + 1) is given as 2π/(1^i).

This highlights for me the direct relationship here as between linear and circular notions (and quantitative and qualitative) in the understanding of the primes!

However because Conventional Mathematics only deals in the quantitative aspects of linear and circular notions (while qualitatively locked in a merely linear 1-dimensional approach) it fails to recognise the true beauty of the primes which relates to this direct link as between quantitative and qualitative notions.

Monday, January 9, 2012

Frequency of Non-Trivial Zeros

As is well known the non-trivial zeros of the Riemann Zeta Function for dimensional value s are given by s = .5 + it (and its conjugate expression s = .5 - it).

Now the first of these zeros of s occurs where the value of t = 14.134725 and the second where t = 21.022040!

The spread (or gap) as between the imaginary part of these first two zeros is approximately 7! However this gap steadily falls as the value of t increases so that when t for example = 100 the average gap is just a little over 2!

In fact a very interesting formula exists to calculate this average spread between zeros (for different values of t) i.e. 2π/log(t/2π) using the natural log based on e.

Thus when t = 100, the average spread = 6.281385/log 15.915494 = 2.27 (approx).


2π represents the circumference of the circle of unit radius.

If we replace in the formula this circular value for the circumference by the corresponding linear value of its radius we obtain 1/log t.


1/log t represents the probability that the number t is prime!

If we obtain the inverse of this expression then log t represents the average gap (or spread) as between primes in the region of t!

So log 100 (= 4.6 approx) for example represents the average spread as between primes in the region of 100.


There appears to be a very close relationship as between both sets of results.

In the first instance the relationship as between real and imaginary values is as between linear and circular. Thus as the spread between non-trivial zeros relates to the imaginary part (of a dimensional value) whereas the corresponding spread as between prime numbers represents a real dimensional value (i.e. the value that e must be raised by to obtain the number t is question) we would expect the relationship as between the two types of spreads in turn to be circular and linear with respect to each other.

Also the relationship as between linear and circular is of a reciprocal nature. For example the prime number 7 is a good example of a linear number (with no sub factors). However its reciprocal 1/7 = .142857 (recurring) is a prime example of a number with a circular structure!


One would expect the spread as between primes and that as between the non-trivial zeros to be the inverse of each other. Thus though the primes thin out in frequency as we ascend the real number scale, the frequency of non-trivial zeros increases as we ascend the imaginary number scale.

What this in fact entails is that as the number of primes thins out, the corresponding number of composite natural numbers proportionately increases (with a greater average number of prime factors).


The connections between both estimates (i.e. for average spread as between non-trivial zeros and the average spread as between primes) is so close that one would seem to imply the other. They are like two sides of the same coin (as it were).

A fascinating corollary of this would imply that just as it is possible to correct the general estimate of prime number distribution with a set of spiral deviations (based on the non-trivial zeros) to get a precise estimate, likewise in principle it should be possible to correct the general estimate of frequency of occurrence of non-trivial zeros (up to a certain number t) with a corresponding set of deviations based on the primes to get the precise number of zeros.

In fact the general estimate of occurence of zeros up to t is given by

N(t) = t/2π*log(t/2π) - t/2π + O(log t)

Ignoring the final error term this would predict that the number of zeros up to t = 100 = 15.9155*2.767293 - 15.9155 = 15.9155*1.767293 = 28.127.

This equates very well for the actual number of zeros = 29.


Recently I have realised that the final O log term which would be used to correct the deviation between predicted and actual zeros could involve a corollary inverse statement to the Riemann Hypothesis in the recognition that all prime numbers are ultimately based on 2!


Incidentally it is interesting when we substitute 2iπ (rather than 2π) in the original formula for calculation of spread between the non trivial zeros.

Then we get 2iπ/log (t/2iπ)

Now 2iπ = log 1.

Therefore we can rewrite formula as log 1/{log(t/log 1)}.

Once again if we replace log 1 with 1, we get 1/log t (i.e. the probability that t is prime).


Also in the latter formula for calculation of frequency of zeros up to the number t (on imaginary scale) if we replace the (circular) circumference of unit circle 2π by its corresponding (linear) radius the RHS of expression

= t(log t) - t = t(log t - 1).

The calculation of frequency of primes up to t is most simply given as t/log t.

However the accuracy of the estimate can be improved for lower t) by substituting log t by log t - 1.

For example if t = 1,000,000 the estimate of the number of primes up to this number
using t/log t = 72,382 (as against true estimate of 78,498)

However if we use t/(log t - 1), we get estimate of 78,030!