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) = π

^{2}/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.

## No comments:

## Post a Comment