Some Interesting Connections

In an earlier entry, I showed a simple way to prove that with respect to the Zeta 1 (Riemann) function,

 _∞

∑{ζ₁(s) – 1} = 1.

 ²

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

ζ₁(2) – 1       = 1/2² + 1/3² + 1/4² +  …       =  .64493…

ζ₁(3) – 1       = 1/2³ + 1/3³ + 1/4³ +  …       =  .20203…

ζ₁(4) – 1       = 1/2⁴ + 1/3⁴ + 1/4⁴  + …        =  .08232…

ζ₁(5) – 1       = 1/2⁵ + 1/3⁵ + 1/4⁵  + …        =  .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. ζ₁(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.