Two Complementary Zeta Functions (5)

Once again it is important to bear in mind that I define two complementary Zeta Functions.

The first relates to the recognised Riemann Zeta Function (Zeta 1) which is defined as an infinite series:

1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + ….

Now if with respect to a number raised to a power we refer to the initial number as the base quantity and the exponent or power as the dimensional number,
for the Zeta 1 Function the natural numbers serve as the base quantities and s as the dimensional power - negative in this case - to which the base quantities are raised.


The Zeta 2 Function - the role of which is largely unrecognised - is defined as a finite series:

1 + s^1 + s^2 + ….+ s^(n – 1).

Here the role of base quantities and dimensional powers is reversed with s now serving as the base quantity which is defined with respect to the natural numbers (from 1 to n) where n has no finite limit.

In a qualified sense, the Zeta 2 can be defined in infinite terms as:

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


It is fascinating to probe the close connections between both Functions.


As is well known with respect to the recognised Riemann Zeta Function (i.e. Zeta 1). (I am inserting 1 and 2 after the ζ sign to distinguish the Zeta 1 and Zeta 2 expressions respectively).


ζ1(1) = 1 + 1/2 + 1/3 + 1/4 + .......

Now each of these terms can be directly related to the Zeta 2 Function.

In terms of this latter Function (using the infinite expression),

ζ2(s) = 1 + s^1 + s^2 + s^3 + s^4 + ...

Now when s = 1/2 with respect to this series,

ζ2(1/2) = 1 + 1/2 + 1/4 + 1/8 + ... = 2

Therefore, s^1 + s^2 + s^3 + s^4 + ... = ζ2(s)- 1 = 1

Then when s = 1/3

ζ2(1/3) = 1 + 1/3 + 1/9 + 1/27 +... = 3/2

Therefore, s^1 + s^2 + s^3 + s^4 + ... = 3/2 - 1 = 1/2

In general s^1 + s^2 + s^3 + s^4 + ... = 1(1 - s)

So ζ2(1/4) = 1/3, ζ2(1/5)= 1/4 and so on!


Again,

ζ(1)1 = 1 + 1/2 + 1/3 + 1/4 + .......

= ∑(s^1 + s^2 + s^3 + s^4 + ... ) where the value of s ranges over the reciprocals of all natural numbers except 1).

and s^1 + s^2 + s^3 + s^4 + ... = ζ2(s) - 1.


Now with respect to the Zeta 1,

ζ1(k) = (1)^k + (1/2)^k + (1/3)^k + (1/4)^k + ...


So for example when k = 2.

ζ1(2) = (1)^2 + (1/2)^2 + (1/3)^2 + (1/4)^2 + ...

= 1 + 1/4 + 1/9 + 1/16 + ....



Therefore in more general terms.

ζ1(k) = ∑(s^1k + s^2k + s^3k + s^4k + ... ),

where s^1k + s^2k + s^3k + s^4k + ... = {ζ2(s)^k} - 1


This relationship holds where the Zeta 1 ranges over real positive dimensional values (≥ 1), and the Zeta 2 over real fractional values for s < 1/2). So each term in the Zeta 1, is directly related to an entire Zeta 2 expression. I have mentioned before how each cardinal number is related (ordinally) to its natural number members. So we have an equivalent relationship here where the Zeta 1 expression is composed of a number of Zeta 2 members. Because of notational difficulties when writing on a web-page, I have produced some accompanying material in text! See Two Zeta Functions (Section 1).