_{ ∞}

∑{ζ

_{1}(s) – 1} = 1.^{2}

So we will now illustrate this for the first few positive integer
values of s (i.e. where s ≥ 2).

ζ

_{1}(2) – 1 = 1/2^{2 }+ 1/3^{2 }+ 1/4^{2 }+ … = .64493…
ζ

_{1}(3) – 1 = 1/2^{3 }+ 1/3^{3 }+ 1/4^{3 }+ … = .20203…
ζ

_{1}(4) – 1 = 1/2^{4 }+ 1/3^{4 }+ 1/4^{4 }+ … = .08232…
ζ

_{1}(5) – 1 = 1/2^{5 }+ 1/3^{5 }+ 1/4^{5 }+ … = .03692…
… … …

… … …

Thus the Zeta 1 expressions here result from reading the
terms here across the respective horizontal rows.

However the corresponding Zeta 2 expressions result from
reading terms down the corresponding vertical columns (representing geometric
series) .

So the 1st column here = 1/4 + 1/8
+ 1/16 + 1/32 + … = 1/2

The 2nd column then =
1/9 + 1/27 +
1/81 + 1/243 + …
= 1/6

The 3

^{rd}column = 1/16 + 1/64 + 1/256 + 1/024 + … = 1/12
So the sums of these vertical columns i.e. 1/2 + 1/6 + 1/12
+ … by definition equals the corresponding sums of the horizontal rows i.e. .64493…
+ .20205… + .08232… +…

And 1/2 + 1/6 + 1/12 + … represents half the values of the
corresponding reciprocals of the triangular numbers i.e. 1 + 1/3 + 1/6 + … = 2.

Therefore the sum of 1/2
+ 1/6 + 1/12 + … = 1.

Likewise therefore the sum of .64493… + .20205… + .08232… + … = 1.

So as well as illustrating the close complementary links as
between the Zeta 1 and Zeta 2 functions (which are horizontal and vertical with
respect to each other), this also helps to prove a very interesting feature
with respect to the sums of the Zeta 1 (Riemann) series (for real integer values
of s ≥ 2)

Now as we have seen the sum of reciprocals of the triangular
numbers is directly associated with the unique number sequence for (x – 1)

^{n}= 0 (where n = 3).
And as we have seen in general terms that the sums of
reciprocals associated with the unique number sequences of (x – 1)

^{n}= 0, = (n – 1)/(n – 2) i.e. 2/1, 3/2, 4/3, 5/4 and so on.
And then when we subtract 1 from each of these values we
obtain 1, 1/2, 1/3, 1/4 …

In other words we obtain the harmonic series, that from one
perspective represents the sum of reciprocals of the unique number sequence associated
with (x – 1)

^{n}= 0 (where n = 2).
Equally from another perspective it represents the Zeta 1 (Riemann)
function i.e. ζ

_{1}(s), where s = 1.
Therefore
we can now perhaps better appreciate the intimate connections as both between the both
The Zeta 1 and Zeta 2 functions and this new alternative function based on the
the sums of reciprocals of the unique number sequences associated with (x
– 1)

^{n}= 0.
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