L-functions et alia

Just as the L1 function can be given two expressions i.e. sum over the positive integers and product over the primes respectively, equally the L2 function can be given two expressions. In previous blog entries I have referred to this latter expression (which has its roots in the L1 function) as the Alt L2 function. Now with respect to the Riemann zeta function, a special case occurs where s = 1, yielding the harmonic series, 1/1 + 1/2   + 1/3 + 1/4 + … This can then be expressed in terms of combinations as 1/1C 0 + 1/2C 1 + 1/3C 2 + 1/4C 3   + … Now, with respect to the denominator, we can define a new series where the nth term represents the sum of the n terms of the previous series. So we thereby obtain 1/1 + 1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 + 2 + 3 + 4) + … = 1 + 1/3 + 1/6 + 1/10 + … Now this can equally be expressed in terms of combinations, i.e. 1/2C 0 + 1/3C 1 + 1/4C 2 + 1/5C 3   + … So the get this new series, we ...

In the last entry, I indicated how each individual term, in the product over primes expression of the L function, can be represented as a complementary L function (where the base and dimensional aspects of number are inverted). I thereby refer to the former collective function as the L1 function and the latter individual function (representing a single term) as the L2 function. So the (collective) L1 function is comprised of an infinite series of (individual) L2 functions. And once again these operate in a dynamic complementary manner with respect to each other. So again in general terms (s > 1), with respect to the Riemann zeta function, the L1 function is customarily represented (in its sum over the positive integers form) as 1 – s + 2 – s + 3 – s + 4 – s   + …, with the corresponding L2 function represented as 1 + (1/p s ) 1  + (1/p s ) 2  + (1/p s ) 3  + (1/p s ) 4  + …, where p is defined over all the primes.   How...

In the last blog entry, I have been concentrating attention on the complementary dynamic relative nature of both the infinite sum over the natural numbers and corresponding infinite sum over the primes expressions (which characterise all L-functions). Therefore what is interpreted in an analytic (quantitative) manner with respect to the right-hand expression (i.e. product over primes) should be interpreted in a corresponding holistic (qualitative) manner with respect to the left-hand expression (i.e. sum over natural numbers). Equally in reverse, what is interpreted in a holistic (qualitative) manner with respect to the right-hand expression should be interpreted in a corresponding analytic (quantitative) manner with respect to the left-hand expression. Thus all individual terms (and collection of terms) with respect to both expressions can thereby be given twin analytic (quantitative) and holistic (qualitative) interpretations that interact with each other in a dynamic c...

A necessary feature of all L-functions is that they can be expressed as infinite expressions, where a sum over the natural numbers is equal to a corresponding product over the primes for real exponents of s (s > 1). For example with respect to the Riemann zeta function (which is the best known) an infinite sum over all the natural numbers is equal to an infinite product over all the primes.      ∞ So ∑1/n s     =       ∏1/(1 – 1/p s )     n = 1                          p So when for example s = 2, 1/1 2 + 1/2 2 + 1/3 2 + 1/4 2 + …        = 4/3 * 8/7 * 25/24 * 49/48 * …    = π 2 /6 However the key significance of this relationship is missed from the conventional mathematical perspective, where the equations are interpreted in...

We saw yesterday how every prime can be given both an independent quantitative identity (as a “building block”) of the natural number system and a shared qualitative identity (as a unique constituent factor of composite natural numbers). And once again this key distinction as between the analytic and holistic nature of primes is unrecognised in conventional mathematical terms with the latter holistic meaning (in every relevant context) reduced in a mere analytic fashion. And this problem by its very nature cannot be remedied within the present accepted mathematical paradigm, which is not geared to deal properly with holistic meaning that is inherently of an unconscious nature (though indirectly capable of expression in a paradoxical rational manner).   So again from the analytic perspective, a prime such as 2 is given an absolute unambiguous meaning in quantitative terms. However when 2 is used as a factor of an even composite number, it then enjoys a relat...

We saw how every number can be equally given a holistic (qualitative) as well as analytic (quantitative) meaning. In direct terms, the latter holistic meaning is equivalent in psycho-spiritual terms to an intuitive energy state. And as psychological and physical aspects of reality are dynamically complementary, this entails that number can equally be given a holistic meaning in terms of physical energy states. Because of the close correspondence as between the Riemann zeros and certain data representing the excited energy states of atomic particles, this clearly suggests that the zeros relate directly to the holistic - rather than analytic - aspect of number. However speaking still somewhat in general terms, a fascinating and crucially important point can be made regarding the holistic nature of number. As we know modern science, with its strong quantitative bias is rooted in analytic interpretation. However what is not all yet realized in our culture is that li...

We have already looked in detail at the number “2” distinguishing clearly both its analytic (quantitative) and holistic (qualitative) aspects of interpretation. So from the former perspective 2 can be viewed in quantitative terms as composed of independent homogeneous units. However from the latter perspective 2 can equally be viewed in qualitative terms as uniquely composed of 1 st and 2 nd units which are interdependent (i.e. interchangeable with each other). And to properly preserve both distinctions, 2 must be viewed in a dynamic interactive manner where quantitative and qualitative aspects are understood as complementary. And what is true above in relation to the number “2” is equally true of all numbers. So rather than being understood in abstract terms as absolute quantitative entities, as for example with the natural numbers - all numbers rightly represent dynamic interacting patterns, entailing the complementary action of both analytic (quantitative) and ...

We saw in the last blog entry how the number 2 can be given both analytic (quantitative) and holistic (qualitative) interpretations that are dynamically interdependent in complementary fashion with each other. And this inherent feature can in principle be extended to all numbers. So far from representing abstract independent entities with an absolute interpretation in quantitative terms, the very nature of number is inherently dynamic and relative, representing the two-way interaction of both its quantitative and qualitative aspects. And the fundamental nature of addition and multiplication is revealed through this new understanding, where these two operations likewise operate in a dynamic complementary fashion enabling understanding to ceaselessly switch as between the quantitative and qualitative type aspects of number.   In a way I find it utterly surprising that this fact is not more widely appreciated for I have been keenly aware from childhood of an obvious prob...