However, this then raises the important question as to
whether we can equally express each of the individual terms of the Zeta 1 (Riemann)
function - both in the sum over natural numbers and product over primes
expressions - in corresponding Zeta 2 terms, as an infinite sum of additive terms!

And remarkably, through our recent investigation of the
unique number sequences associated with the general polynomial expression i.e. (x
– 1)

^{ n}= 0, this can now be made possible.
To make this a little easier to illustrate, I will start with
the product over primes expression for ζ

_{1}(s), where s = 1.
Thus ζ

_{1}(1) = 2/1 * 3/2 * 5/4 * 7/6 * …
Of course this expression which equates to the harmonic
series (in the sum over natural numbers expression) does not converge to a finite answer!

However we can still equate each individual term with an
infinite series based of the sum of the reciprocals of the unique digit
sequences associated with (x – 1)

^{n}= 0.
So when n = 3, the unique digit sequence associated with (x
– 1)

^{3}= 0 is
1, 3, 6, 10, 15, 21, …

Thus the infinite series of the sum of the reciprocals of
these numbers

= 1 + 1/3 + 1/6 + 1/10 + 1/15 + … = 2/1

And this in turn represents the 1

^{st}term of the Zeta 1 (Riemann) product expression over the primes for s = 1.
Now it is easier to demonstrate that our new reciprocal
expression does in fact represent a sum over all the natural numbers in the
following manner.

So 1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + …

= 1/1 + 1/(1 +2) + 1/(1 + 2 + 3) + 1/(1+ 2 + 3 + 4) + 1/(1 +
2 + 3 + 4 + 5) + …

Thus in general terms the denominator of the nth term
represents the sum of the first n natural number terms!

And further reciprocal expressions with respect to the
unique digit expressions of

(x – 1)

^{n}= 0 for (n > 3) involve in their denominators, compound combinations involving all the natural numbers (up to n).
For example, the 2

^{nd}term in the product expression for ζ_{1}(1) = 3/2.
Now this in turn equates with infinite sum of the
reciprocals associated with the unique number sequence for (x – 1)

^{4}= 0, i.e.
1 + 1/4 + 1/10 + 1/20 + 1/35 + … = 3/2.

And the denominators of any term t, represents compound expressions entailing all the natural numbers to t.

For example the 3

^{rd}term = 1/10.
And the denominator 10 = 1 + (1 + 2) + (1 + 2 + 3)!

Finally, to fully illustrate this point, the 3rd term in the product expression for ζ

_{1}(1) = 5/4
This in turn equates with infinite sum of the reciprocals
associated with the unique number sequence associated with (x – 1)

^{6}= 0, i.e.
1 + 1/6 + 1/21 + 1/56 + 1/121 + … = 5/4.

The denominator of the 2

^{nd}term - which in this case is easiest to illustrate - then represents a compound expression entailing the first two natural numbers i.e.
1 + {1 + [1 + (1 + 2)]}.

Though we have illustrated here with respect to the Zeta 1
(Riemann) product expression over all the primes for s = 1, corresponding further
reciprocal expressions, based on the unique number sequences associated with (x
– 1)

^{n}= 0 can be found for all Zeta 1 product expressions where s is an integer > 1.
We can likewise associate each of the individual terms in
the sum over natural numbers Zeta 1 (Riemann) expressions with infinite series based on
the reciprocals of the unique digit sequences associated with (x – 1)

^{n}= 0.
Again for example in the simplest case where s = 1.

ζ

_{1}(1) = 1 + 1/2 + 1/3 + 1/4 + … (the harmonic series)
Now the 1

^{st}term here can be expressed through the infinite reciprocal sequence - already considered - associated with (x – 1)^{3}= 0.
So 1 = (1 + 1/3 + 1/6 + 1/10 + …) – 1.

Then the 2

^{nd}term 1/2 can be expressed through the infinite reciprocal sequence associated with
(x – 1)

^{4}= 0 i.e.
1/2 = (1 + 1/4 + 1/10
+ 1/20 + …) – 1.

Then the 3rd term 1/3 can be expressed through the infinite
reciprocal sequence associated with

(x – 1)

^{5}= 0 i.e.
1/3 = (1+ 1/5 + 1/15 + 1/35 + …) – 1.

And we can continue on indefinitely in this manner with all
further terms for real integer values of

ζ

ζ

_{1}(s), where s > 1.
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