In the past two blogs I have been addressing the nature of the Riemann zeta function for s = 0.

However strictly, as in dynamic interactive terms two complementary functions i.e. Zeta 1 and Zeta 2 always interact, I have been addressing the nature of the Zeta 1 function i.e. ζ(s

_{1}), where s

_{1}= 0.

And just as an important functional equation connects ζ(s

_{1}) and ζ(1 – s

_{1}), likewise in corresponding manner, an important, though very simple, functional equation connects ζ(s

_{2}) and - in this case -

ζ

_{2}(1/s

_{2}), where ζ(s

_{2}) = 1 – ζ(1/s

_{2}).

Now, again as we have seen, the one value for which ζ(s

_{2}) is undefined is where s

_{2}= 1. However this implies also that 1/s

_{2 }= 1 with both sides of the functional equation remaining undefined in this limiting case.

And we have also seen in yesterday's entry, how both the Zeta 1 and Zeta 2 functions are connected through the corresponding Eta 1 (related to the Zeta 1 function).

So, ζ(s

_{1}) = η(s

_{1})/(1 – s

_{2}). And as ζ(s

_{2}) = 1/(1 – s

_{2}), this implies that,

ζ(s

_{1}) = η(s

_{1}) * ζ(s

_{2}).

Alternatively we can say that,

ζ(s

_{2}) = ζ(s

_{1})/η(s

_{1}).

We now move on to the intriguing case of the Riemann zeta function where s (i.e. s

_{1}) = – 1.

So ζ(s

_{1}) = 1/1

^{– 1 }+ 1/2

^{– 1 }

^{ }+ 1/3

^{– 1}

^{ }+ 1/4

^{– 1 }

^{ }+ ...

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

So this represents the sum of the natural numbers, which in the standard linear manner of interpretation, diverges to ∞.

However through analytic continuation, the Riemann zeta function can be extended from "intuitively meaningful" results for s > 1 to "non-intuitive" results for negative values < 0 (including the limiting value of s = 0).

However in truth all numbers possess both a quantitative aspect, relating to their independent and a qualitative aspect, relating to their corresponding interdependent identity.

Numbers therefore must be appropriately interpreted in a dynamic interactive manner, where both quantitative and qualitative aspects are understood in a relative complementary manner.

So results of the Riemann zeta function appear as "intuitively meaningful" for values of s > 1, because these concur with the standard quantitative (or analytic) aspect of number interpretation.

However by the same token, results of the Riemann zeta function appear as non-intuitive (for s ≤ 0) precisely because these do not now concur with the standard quantitative interpretation.

In others words, the meaning of these results is of a direct qualitative (or holistic) nature, which is related to the unconscious - rather than the conscious - aspect of understanding.

However, we can indirectly attempt to convey what is implied by the value of ζ(s

_{1}) = – 1/12.

Once again we start by attempting to obtain the corresponding Eta 1 expression for,

ζ( – 1) = 1 + 2 + 3 + 4 + ...

So η( – 1) = 1 – 2 + 3 – 4 + ...

One simple approach, which does not require even simple differentiation, is to start by grouping the terms in pairs.

Therefore

η(– 1) = (1 –
2) + (3 – 4) + (5 –
6) + …

= – 1 – 1 – 1 – …

Now we have already seen
that ζ(0) = 1 + 1
+ 1 + … = – 1/2.

This
might suggest therefore that η(– 1) = – (– 1/2) = 1/2.

However
because of the pairing of terms with respect to η(– 1),
this means that it contains only half the terms contained in ζ0).

Therefore,
η(–
1) = 1/2 * 1/2 = 1/4.

One
can just about intuitively accept - in terms of conventional linear interpretation
- that η(0) = 1 – 1 + 1 – … = 1/2.

However
in then “converting” this value to obtain the corresponding value for ζ(0) =

– 1/2, the intuitive nature of the new result is
lost.

This
is due to the fact - as I explained in my last entry - that a proper holistic interpretation
is now required.

So
what is interesting here is that in obtaining ζ(– 1) we
have to now “convert” a value i.e. η(– 1) that is already
non-intuitive.

So in
a very real sense, the ultimate value obtained for ζ(–
1) = – 1/12 is thereby “doubly non-intuitive” from the
conventional linear quantitative perspective.

Therefore
the very important point that needs to be stressed is that that an increasingly
more refined holistic ability is required to properly interpret the numerical results
for ζ(1 – s), where the value of s (representing a positive
even integer) continually increases.

So in
the case of ζ(– 1), s = 2 which is still
relatively simple to interpret in a refined holistic manner. However for ζ(–
3), ζ(– 5), ζ(– 7), … where s = 4, 6, 8, … true
holistic interpretation becomes progressively more demanding.

This
is something that is not at all yet appreciated in present Mathematics. There
are really two forms of interpretation - analytic and holistic - respectively,
that properly apply to all relationships. Because for millennia now Mathematics
has undergone development largely in just one direction, considerable specialisation
has now taken place with respect to its analytic aspect! However its holistic
aspect - which in truth is equally important - is not yet formally recognised
by the profession!

So
quite understandably, true holistic development remains remarkably undeveloped,
which explains why the results of the Riemann zeta function - for the simplest of
negative (odd) integers - appear so “non-intuitive”.
Again appropriate intuition based on true holistic - rather than conventional
analytic - understanding is required.

As we
have seen, ζ(s

_{1}) = η(s_{1}) * ζ(s_{2}).
Thus we “convert” the result for η(–
1) to obtain ζ(– 1) for the Zeta 1 function by
multiplying by ζ(s

_{2}) = 1/(1 – 1/2^{s – 1}).
When s
= – 1, ζ(s

_{2}) = 1/(1 – 4) = – 1/3.
Therefore
ζ(– 1) = 1/4
* (– 1/3) = – 1/12.

s

_{2}= 1/2^{s – 1}= 1/(1/4) = 4 (when s = – 1).^{ }
So ζ(s

_{2}) = 1 + 4 + 16 + … = – 1/3
Again
let us imagine a cake divided into 4 portions, where the first portion chosen
(= 1 slice) represents 1/4 of total cake.

Then
with respect to another of the four starting portions take a quarter portion

(= 1/16
of total cake) and divide it into 4 slices.

Then
with respect to one of the remaining (latest) portions again divide into 4
new portions (with each = 1/64 of total cake) and divide one of these into 16
slices.

So
when one sums up the (part) fractions of the cake, we get,

1/4 +
1/16 + 1/64 + … = 1/3.

However
when we now alternatively add up the whole slices, we get,

1 + 4
+ 16 + … = – 1/3.

Now
the first answer literally represents the (conscious) quantitative manner
of understanding number relationships, where the whole is viewed in terms of
its related parts.

The
second answer represents the (unconscious) qualitative manner of understanding number
relationships where the part is now viewed in terms of its related wholes. So the very nature of addition is now distinct!

And
because unconscious recognition represents the (dynamic) negation of consciously
posited reality, the result carries a negative sign.

Thus
again, it takes a true holistic way of appreciating number relationships to
make this latter result intuitively accessible.

However
in the next entry we will look in more detail at what the important result for

ζ(– 1) properly entails.