The Startling (Unrecognised) Significance of the Riemann Functional Equation

The Riemann Functional Equation establishes an important relationship as between positive and negative values of s for the Zeta 1 Function.

It can be expressed in several ways.

One that I especially like is that given in John Derbyshire's "Prime Obsession" on P. 147:

ζ(1-s) = 2^(1-s)π^(-s)sin{[(1-s)/2]π}(s-1)!ζ(s).


Thus, if we calculate a result for the Function with respect to a positive value of s on the RHS of the Function, we can equally calculate a corresponding value with respect to the negative value of s, i.e. 1 - s.

So for example if we have already calculated ζ(2), then through the formula we can equally calculate ζ(- 1).


However a very significant problem arises with respect to the Function for negative values of s, in that the results seem somewhat meaningless from a conventional linear perspective (which properly constitutes Type 1 Mathematics).


For example the Function for s = - 1, generates the sum of the natural numbers

1 + 2 + 3 + 4 +........

Now in conventional (Type 1) terms this series clearly diverges to infinity.

However the result as derived from the Functional Equation = - 1/12.

We will give a coherent interpretation of what this numerical result really signifies in a future entry. But firstly we have to place it within its proper general context.

Let us return to the Zeta 1 and Zeta 2 formulations of the Function, corresponding to the (linear) Type 1 and the (circular) Type 2 aspects of Mathematics respectively.


So Zeta 1 can be expressed as the infinite series

1/1^s + 1/2^s + 1/3^s + 1/4^s +.......

Now if we invert each term of this Function we then get

1^(- s) + 2^(- s) + 3^(- s) + 4^(- s) +......


Zeta 2 by comparison is expressed by the inverse series (with respect to both base and dimensional numbers as

1 + s^1 + s^2 + s^3 + s^4 + .....

Now when we let s = 0 (as dimensional power) in the Zeta 1 and s = 1 (as base quantity) in the Zeta 2, both Functions appear as identical.

So for s = 0, Zeta 1 gives

1^0 + 2^0 + 3^0 + 4^0 + .....

= 1 + 1 + 1 + 1 + ....


Likewise for s = 1, in Zeta 2, we have

1 + 1^1 + 1^2 + 1^3 + 1^4 +......

= 1 + 1 + 1 + 1 + .....


So in this special case where both Zeta 1 and Zeta 2 appear identical, s = 0 (as dimension) in the former case and s = 1 (as base quantity) in the latter.

Therefore Zeta 2 is connected with Zeta 1 through the simple transformation s = 1 - s (where it is understood that s now switches as between dimensional power and base quantity).


Now to distinguish both Functions, let

Zeta 1 = ζ(s)1 and Zeta 2 = ζ(s)2.

Then in this special special case we have ζ(s)2 = (1 - s)ζ(s)1. So the Functional Equation established by Riemann (with its adjustment terms) can be used to extend this result for all other values of s (but now properly incorporating both Zeta 1 and Zeta 2).


Thus if we interpret - as is the convention - the Functional Equation using solely Zeta 1 where s = 0, then all the non zeta terms cancel out and we are left,

ζ(s) = (1 - s)ζ(s), i.e. the trivial identity

ζ(s) = ζ(s).


Now the important point I am making is that appropriate interpretation of the Riemann Functional Equation requires appreciation that both sides of the equation actually point to two Functions (Zeta 1 and Zeta 2 respectively) that bear a complementary relationship with each other.


So when we interpret quantitative symbols and numerical results in a linear fashion on the RHS in accordance with Zeta 1, this implies that corresponding appropriate quantitative interpretation on the LHS requires a complementary circular interpretation in accordance with Type 2.


Likewise when we give symbols a qualitative circular type interpretation, on the RHS in accordance with Zeta 1 (which I have already commenced in the Riemann Function blogs) this implies that when we come to the LHS, this now implies a complementary quantitative interpretation.


So the startling truth regarding the Riemann Zeta Function, is that its proper appreciation requires incorporation of two Functions, Zeta 1 and Zeta 2, which then operate in a perfect complementary manner with respect to interpretation on both sides of the Functional Equation.

Though the Functional Equation associated with Riemann (and perhaps to a degree Euler) is a marvellous discovery in its own right, unfortunately it completely masks the true nature of the Zeta Function (and of course its associated Riemann Hypothesis) by attempting to convey it merely in quantitative terms, from a Type 1 mathematical perspective. Though it is certainly valid from this context, it leaves the essential nature of both the Function and Hypothesis completely unexplained.


For when we understand correctly, we realise that the Functional Equation (incorporating both Zeta 1 and Zeta Functions) establishes a perfect complementary relationship as between both quantitative and qualitative type meaning (which is inherent in the very nature of the primes).

So, quantitative interpretation with respect to the RHS (in accordance with Zeta 1) is matched by corresponding qualitative interpretation on the LHS (in accordance with Zeta 2).

Likewise qualitative interpretation on the RHS (in accordance with Zeta 1) is matched by quantitative interpretation on the LHS (in accordance with Zeta 2).

And we could equally apply Zeta 2 to interpretation of the RHS with then complementary interpretation through Zeta 1 on the LHS.


So when we read the existing Riemann Function appropriately, we realise that through switching from from RHS (using Zeta 1) to LHS, that both the base quantities and dimensional powers (as defined on the RHS), thereby likewise switch with each other, so that in fact we are now interpreting in accordance with the complementary Zeta 2 formulation!


And if this seems a lot to take in, well its true significance is that what we presently define as Mathematics is simply not fit for purpose.


Using my own terminology, Conventional Mathematics is still rooted in an absolute version of the Type 1 (quantitative) approach.

But Mathematics equally needs a Type 2 aspect in a direct qualitative interpretation of the nature of mathematical symbols. And only then can a truly comprehensive Type 3 approach emerge, where both Type 1 and Type 2 are now understood in dynamic interactive terms as fully complementary with each other.

And if ever a problem needed this Type 3 approach for proper comprehension, the Riemann Zeta Function (and associated Hypothesis) certainly qualifies - pardon the pun - as a prime candidate!