L-functions et alia

It struck me clearly the other day that each L2 function can be expressed as the quotient of two L1 functions. For example in the Riemann zeta function (where s = 2) the first term in the product over primes expression is 4/3, which is related to the corresponding first prime 2, through the relationship 1 – 1/ 2 2 . Now this can be expressed as the quotient of the functions, 1/1 2 + 1/2 2 + 1/3 2   + 1/5 2 + … and 1/1 2   + 1/3 2 + 1/5 2   + 1/7 2 + … (where in the latter case all numbers which contain 2 as a shared factor are omitted). So (1/1 2 + 1/2 2 + 1/3 2   + 1/5 2 + …)/(1/1 2   + 1/3 2 + 1/5 2   + 1/7 2 + …) = 1 + ( 1/2 2 ) 1 + ( 1/2 2 ) 2 + ( 1/2 2 ) 3 + …     = 4/3 And notice the complementarity here! Whereas the natural numbers represent the base aspect in the L1, they represent the dimensional aspect of number in the L2; and whereas 2 is constant as a dimensional number in the L1 it is constant (as reciprocal) in L2. So this properly signifies

As is well known when s = 4, ζ(4) = 1/1 4 + 1/2 4   + 1/3 4 + 1/4 4 + …    = π 4 /90 and when s = 2, ζ(2) = 1/1 2 + 1/2 2   + 1/3 2 + 1/4 2 + …     = π 2 /6, so that (1/1 2 + 1/2 2   + 1/3 2 + 1/4 2 + …) 2   = π 4 /36    Therefore (1/1 2 + 1/2 2   + 1/3 2 + 1/4 2 + …) 2 /(1/1 4 + 1/2 4   + 1/3 4 + 1/4 4 + …)   = 90/36 = 5/2 However we have already seen that this can be written as a Dirichlet L-series i.e. 1/1 2 + 2/2 2 + 2/3 2 + 2/4 2 + 2/5 2 + 4/6 2 + 2/7 2 + 2/8 2 + 2/9 2 + 4/10 2 + … = 5/2 So ζ(2) 2 / ζ(4) = 1/1 2 + 2/2 2 + 2/3 2 + 2/4 2 + 2/5 2 + 4/6 2 + 2/7 2 + 2/8 2 + 2/9 2 + 4/10 2 + … Thus we have expressed the square of one Dirichlet series (the Riemann zeta function where s = 2) divided by another Dirichlet series (the Riemann zeta function where s = 4) by yet another Dirichlet series (where s = 2). And this relationship can equally be expressed as a product over primes expression. So ζ(2) 2 / ζ(4)   = (4/3 *

In the last entry, I mentioned the new general formula, which can be used to extend relationships as between sum over integers and product over primes expressions with respect to the Riemann zeta function i.e. 1/1 s + (1 + k 1 )/2 s + (1 + k 2 )/3 s + (1 + k 1 )/4 s + (1 + k 3 )/5 s + {(1 + k 1 )(1 + k 2 )}/6 s + … = 1/{1 – (1 + k 1 )/(2 s + k 1 )} * 1/{1 – (1 + k 2 )/(3 s + k 2 )} * 1/{1 – (1 + k 3 )/(5 s + k 3 )} * …,   where k 1 , k 2 , k 3 , … are rational numbers which can be either positive or negative. In that entry, I demonstrated the - surely - interesting fact that when s is 2, 4, 6, … and k 1 = k 2 = k 3, …   = 1, that the value of the two expressions (sum over the integers and product over the primes respectively) is a rational number. In this case there is a clear relationship as between the values of the standard zeta function for s and 2s respectively. So - again when s is a positive even integer - the value of the zeta expression (for both sum

It just struck recently that the sum over the integers and product over the primes expressions, with respect to the Riemann zeta function, represent a special case of a more comprehensive relationship, which can be expressed in general terms as follows: 1/1 s + (1 + k 1 )/2 s + (1 + k 2 )/3 s + (1 + k 1 )/4 s + (1 + k 3 )/5 s + {(1 + k 1 )(1 + k 2 )}/6 s + … = 1/{1 – (1 + k 1 )/(2 s + k 1 )} * 1/{1 – (1 + k 2 )/(3 s + k 2 )} * 1/{1 – (1 + k 3 )/(5 s + k 3 )} * …   where k 1 , k 2 , k 3 , … are rational numbers which can be either positive or negative. So the standard equation represents the situation where the values for  k 1 , k 2 , k 3 , …   = 0 .   It can be seen here it is the distinct prime factors of a number that determines the nature of the sum over the integers expression. So for example, 6 is the 1st composite natural number with two distinct prime factors. Therefore as we can see from the new formula we must here thereby multiply 1 + k 1

In general terms with reference to the prototype Riemann zeta function, if we choose the primes (in the product over primes expression) using any infinite ordered sequence of integers, then a corresponding sum over the natural numbers expression can thereby be provided. For example the (infinite) series of triangular numbers i.e. 1, 3, 6, 10, 15, …, provides one such ordered sequence (where we are always enabled to provide the next term in the sequence i.e. n(n + 1)/2, with n = 1, 2, 3, … So if we now apply this ordering to the primes by choosing the 1 st , 3 rd , 6 th , 10 th 15 th … numbers, then we have 2, 5, 13, 29, 47, … So the corresponding sum over the integers expression is thereby now based (including 1) on using these primes as sole factors. Thus 1/1 s + 1/2 s +1/4 s + 1/5 s + 1/8 s + 1/10 s …   = 1/(1 – 1/2 s ) * 1/(1 – 1/5 s ) * 1/(1 – 1/13 s ) * … In fact the prototype Riemann zeta function itself corresponds on an ordering that is equiva

 In yesterday's blog entry I dealt with the notion of - what I refer to as - the different orders of primes and natural numbers respectively. So potentially an unlimited number of orders with respect to both the primes and corresponding natural numbers exist, defined as Order k primes and Order k natural numbers respectively. And in each case the Order k natural numbers can be represented as a sum over the integers, which can then be matched in corresponding fashion by a corresponding product over the Order k primes. So again the Order 1 natural numbers (representing all the natural numbers) can be matched with the Order 1 primes (representing all the primes). Thus again, 1/1 s + 1/2 s + 1/3 s + 1/4 s + …     = 1/(1 – 1/2 s ) * 1/(1 – 1/3 s ) * 1/(1 – 1/5 s ) * … Then the Order 2 natural numbers (representing the products of prime factors which have themselves an ordinal prime ranking) can be matched with the Order 2 primes (i.e. as the primes with an ord

I have raised before the notion of different primes with respect to the natural number system. The well-known set of primes 2, 3, 5, 7, 11, … in this context can be referred to as Order 1 primes, with the corresponding natural numbers 1, 2, 3, 4, 5, …, in corresponding fashion referred to as Order 1 natural numbers. However we can now order the cardinal primes 2, 3, 5, 7, 11, …, with respect to ordinal natural number rankings 1, 2, 3, 4, 5, … If we now choose to list only those primes with a corresponding ordinal prime ranking we now obtain a new set of cardinal primes i.e. 3, 5, 11, 17, 31, 41, … I then refer to this new set of cardinal primes as Order 2 primes with the corresponding set of natural numbers (including a) based on the use of these primes as factors as 1, 3, 5, 9, 11, 15, 17, 25, 27, 31, … And I refer to this new set of natural numbers as Order 2 natural numbers. We can now again give the Ordinal 2 cardinal primes an (Order 1) natural number

We have in the last few entries emphasised the dynamic complementary nature of the Riemann zeta function - which by extension applies to all L-functions – in the manner in which both the analytic (quantitative) and holistic (qualitative) nature of number interacts in two-way fashion with each other. This also implies a corresponding dynamic complementarity as between both L1 and L2 functions. As we have seen whereas the collective whole function (with respect to both RHS product over primes and LHS sum over the integers expressions) is expressed as an L1 function, then each individual term is then expressed as a corresponding L2 function. And just as we have seen the collective whole L1 function can be given two alternative expressions, likewise each individual L2 function can likewise be given two alternative expressions. So again with respect to our prototype L function i.e. the Riemann zeta function   ∞                       ∞  ∑ 1/n s    = ∏1/(1 – 1/p s )

As is well-known when s is a positive even integer, the value of ζ(s) can be represented in the form kπ s (where k is a rational fraction). Thus again in the best known case where s = 2 ζ(2) = 1/1 2 + 1/2 2 + 1/3 2 + 1/4 2 + …   = 4/3 * 9/8 * 25/24 * 49/48 * …    = π 2 /6 And as we have seen the general form of this relationship i.e. kπ s will continue to hold when we eliminate on the RHS (product over primes expression) an individual term or series of terms (based on a particular prime or series of primes) and then in corresponding fashion eliminate with respect to the LHS (sum over integers expression) the collection of terms where this prime or series of primes operate(s) as shared common factor(s). Now this is already known since the time of Euler from an analytic (quantitative) perspective. However what is fascinating - and indeed very important - is the fact that a convincing holistic (qualitative) mathematical rationale can likewise be given for the rela

In yesterday’s blog entry, I illustrated the relationship as between each individual term (in the product over primes expression) with the collective nature of terms (in the corresponding sum over natural numbers expression). And the crucial point here is that from the appropriate dynamic perspective that a complementary relationship exists so that what is interpreted in an analytic (quantitative) manner with respect to each individual term, related to a specific prime, is then interpreted in a holistic (qualitative) manner with respect to the collection of terms where this specific prime operates as a shared common factor. However as always with such dynamic complementary relationships reference frames can be switched so that equally each individual term (with respect to the product over primes expression) can be viewed in a holistic (qualitative) manner. Then when this is the case the corresponding interpretation with respect to the collection of natural numbers where the

Once again for the Riemann zeta function - which serves as the prototype for all L-functions - can be expressed both as a sum over all the positive integers and a product over all the primes. So when s is an integer (> 1), both expressions converge with     ∞                       ∞   ∑ 1/n s    = ∏1/(1 – 1/p s ) n = 1                 p = 2 Thus in the well-known case, where s = 2, 1/1 2 + 1/2 2 + 1/3 2 + 1/4 2 + …       = 1/(1 – 1/2 2 ) * 1/(1 – 1/3 2 ) * 1/(1 – 1/5 2 ) * … i.e. 1 + 1/4 +   1/9 + 1/16 + …      = 4/3 * 9/8 * 25/24 * …      = π 2 /6 Now each (individual) prime-based term on the RHS expression is linked to the corresponding (collective) natural number expression in a precise manner where dynamic complementarity as between analytic (quantitative) and holistic (qualitative) is key in terms of appropriate interpretation. So the first (individual) term in the RHS expression i.e. 4/3 is related in turn to first prime i.e 2. Now when we give