New Approach to L-functions

This  is carrying on from previous entries to the blog on the Riemann Hypothesis where I have firmly concluded that a radically distinctive mathematical approach is required to properly understand the very nature of the Riemann zeta function and its associated Riemann Hypothesis.

And as the Riemann zeta function, which is the most famous, represents but a special case of what are referred to as L-functions, then the same radically distinctive approach is thereby required to understand this ubiquitous phenomenon.

In fact there is a web-site LMFDB i.e. “The L-functions and Modular Forms Database” that provides a wonderful resource of extensive information on some 20 million L-functions, which are related to various classes of mathematical objects of increasing complexity.

The Riemann zeta function represents the central example of one - relatively simple - class, known as Dirichlet L-functions (based on the field of rational numbers).  However there are other types related to more generalised algebraic number fields (i.e. Dedekind zeta functions) that give rise to their own L-series with similar properties to the Riemann zeta function function.

Then there other classes of L-functions, related to algebraic varieties such as elliptical curves.

L-functions depending on their complexity are classified according to various degrees from 1-4 (with the simpler Dirichlet functions of degree 1).

There is also a distinction as between algebraic and transcendental L-functions with the former the solution of algebraic equations (in a finite amount of steps) whereas the latter properly require an infinite procedure so that values can only be approximated.

An announcement in 2008 related to the discovery of the first example of a 3^rd degree transcendental L-function which required advanced theoretical discoveries and 10,000 hours of computer time. So this increasingly specialised abstruse area is certainly not for the faint-hearted!

However it's the features that all L-functions have in common that I wish to deal with in this blog indicating that all rightly require the same dynamic means of interpretation (which is not yet formally recognised by the mathematical profession).

Now these common features can be conveniently summarised as follows.

1) All L-functions can be given both as a Dirichlet series sum over natural numbers and as a Euler type product over the primes.

For example the Riemann zeta functions can be expressed both as a Dirchlet series and Euler product as follows;

ζ(s) = 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) where Re(s) > 1.

Thus the LHS of the expression entails an infinite sum over all the natural numbers, while the RHS entails a corresponding infinite product over all the primes.

And this is a universal feature of all L-functions where an infinite sum entailing natural numbers matches a corresponding product over primes.

2) The function that holds for Re(s) > 1 can be analytically continued to the rest of the complex plane (except perhaps for a very limited number of values).

Where this is the case a functional equation can be derived that maps values of function for s on the RHS of the plane to corresponding values for 1 – s on the LHS.

So again for example, with the Riemann zeta function when s = 2, ζ(2) = π²/6.

The corresponding value then for  ζ(1– s), i.e ζ(– 1), established through the functional equation,  = – 1/12.

3) Each L-function has its own Riemann Hypothesis with the imaginary part of its zeros (for s) all postulated to lie on the critical line drawn vertically through 1/2.

The extended Riemann Hypothesis is the conjecture that Dedekind zeta functions have all zeros on the critical line (through 1/2).

The generalised Riemann Hypothesis is the conjecture that all Dirichlet zeta functions - or as sometimes stated globally, the entire range of L-functions - have their zeros on the critical line.

4) L-functions encode in various ways the nature of the distribution of varying configurations of primes with respect to corresponding configurations of the natural numbers.

The Riemann Hypothesis provides the most universal - and indeed most important - example of this distribution relating to that of all the primes and natural numbers.

However one simple example of an alternative Dirichlet L-function can be obtained by dividing each number by 4 (ignoring all even terms) with terms leaving a reminder of 1 given a positive sign and terms leaving a remainder of 3 a corresponding negative sign.

This L-function then encodes knowledge of the distribution of the class of primes (leaving a remainder of 1 when divided by 4) among the odd natural numbers.     

However perhaps surprisingly, I would strongly contend that the most important shared feature of all is at present entirely overlooked by the mathematical profession.

Now I will be developing this point in detail in future blog entries. However it would be perhaps appropriate to introduce my position here.

All mathematical symbols - when rightfully interpreted - can be given both analytic and holistic meanings.

Analytic in this context is synonymous with the standard mathematical interpretation of symbols, where the quantitative aspect (especially in relation to number) is solely emphasised.

So for example with respect to the primes - and indeed natural numbers - in conventional mathematical terms, these are unambiguously understood as representing number quantities.

Now, as we know, analytic is given a more specialised meaning in Mathematics as for example in reference to analytic functions (containing an infinite number of terms).

However I typically use the term in a much broader sense to refer to the standard mathematical practice, whereby the whole in any context is quantitatively reduced to its component parts.  So in this context, for example, when one maintains that 2 = 1 + 1, the “whole” number 2 is expressed as the quantitative sum of its part members (1 and 1).

Thus though analytic - in the sense that I customarily employ - necessarily includes the more specialised mathematical definition of analytic, it conveys a much wider meaning (referring universally to present accepted mathematical interpretation).

Remarkably however, all mathematical symbols can be equally given an important holistic meaning.

Holistic in this context refers to the qualitative aspect of interpretation, which is not formally recognised by the mathematical profession.

Just stop for a moment and attempt to grasp what is being said here (without any attempt at hyperbole)!

There is an equally important qualitative aspect to the interpretation of all mathematical symbols which at present is entirely ignored - certainly in formal terms - by the mathematical community.

Furthermore both analytic (quantitative) and holistic (qualitative) aspects are properly designed to be used in a dynamic interactive manner (where they are understood in complementary fashion).

Though admittedly enormous mathematical progress has been made in recent years through the continued development of the analytic (quantitative) aspect, at best it can only provide a limited and - crucially - unbalanced appreciation of mathematical reality.

And this is especially the case where fundamental understanding of the very nature of number is required.

So the most important common attribute of all L-functions is the yet - unrecognised fact - that they incorporate in a truly amazing manner the dynamic complementary interaction of intimately related analytic (quantitative) and holistic (qualitative) aspects of number relationships.