Enhanced Zeta 2 Functions (3)

As we have seen from the last entry, all the terms in the product over primes expression (s ≥ 1) for the Zeta 1 (Riemann) function can be directly constructed from the corresponding Zeta 2 function,

i.e. 1 + s₂¹ + s₂² + s₂³ + .... where s₂ = 1/p, with p ranging over all the primes.

From the opposite perspective it is likewise possible to directly generate all the individual Zeta 1 functions - for 1/p with again p ranging over all the primes - in the sum over natural (dimensional) numbers directly from the Zeta 1 (Riemann) function. This is is done by taking the  Zeta 1 function (i.e. sum over natural numbers) for each of the integer values of s ≥ 1, written as horizontal rows, and then reading each of the vertical columns that arise, as corresponding Zeta 2 expressions (for the various values of s) .

Thus - rather like electricity and magnetism in physics - both functions really represent but different perspectives on the same underlying bi-directional relationship as between the primes and natural numbers in both cardinal (quantitative) and ordinal (qualitative) terms.

The Zeta 1 (Riemann) function is directly concerned collectively with the (external) macro relationship of the primes with the natural numbers as a whole; the Zeta 2 function then in corresponding fashion, is directly concerned individually with the (internal) micro relationship of prime and natural factors (within each number).

And just as macro and micro reality are inextricably linked at a physical level, in a more fundamental manner, the macro and micro aspects of the relationships between primes and natural numbers both as (base) numbers and (dimensional) factors are inextricably linked.

Now as we have seen, in general the Zeta 1 connection as between sum over natural numbers and product over primes can be simply given as

 ∑ 1/n^s    =    ∏ 1/(1 – 1/p^s)  and in these explorations we have been so far concentrating on the situation where s ≥ 1.

However there is a related product over primes expression that is given as ∏ 1/(1 + 1/p^s).

In fact, the appreciation of the nature of these two related products for the product over primes, provides the explanation as to the decidedly different structure of the Zeta 1 (Riemann) function for even and odd positive integer values of s respectively.

However we will initially return to the Zeta 1 function to look at  its product over primes expression, when s = 1.

Now again this relates (in the sum over natural numbers) to the well known harmonic series (which diverges).

However it can be given a corresponding product over primes expression, which, of course equally diverges.

This product expression for ∏ 1/(1 – 1/p^s), where s = 1,

= 2/1 * 3/2 * 5/4 * 7/6 * ...

However there is a corresponding (shadow) product expression for ∏ 1/(1 + 1/p^s), where s = 1,

= 2/3 * 3/4 * 5/6 * 7/8 * ...

Now this is deeply relevant in understanding the structure of ∏ 1/(1 – 1/p^s), where s = 2, for this can now be expressed as

∏ {1/(1 – 1/p¹) * (1 + 1/p¹)}

In other words ∏ 1/(1 – 1/p²) is obtained from multiplying together (separately for each term) each of the two previous product terms (where s = 1) before then multiplying together the combined terms.

So ∏ 1/(1 – 1/p^s) = 4/3 * 9/8 * 25/24 *49/48 *...

And as we can see each term represents a product of the two complementary terms ,

i.e. 1/(1 – 1/p¹) * (1 + 1/p¹) respectively.

So for example, the 1st term (in the product expression (for s = 2) = 2/1 * 2/3 = 4/3 and so on in like manner for all the other terms.


And in general terms, this is true for all values of the function where s represents a positive even integer.

For in the general case where s = 2 k and thereby an even no. with k ranging over the natural number integers 1, 2, 3,.....  etc., then, 

∏ 1/(1 – 1/p^s) = ∏ {1/(1 – 1/p^k) * (1 + 1/p^k)}, with k a positive integer = s/2.


So there is complementarity in evidence here as between the negative and positive signs with respect to 1/p^k in the formula.

However where s is an odd even integer, there is no way of expressing the product over primes

∏ 1/(1 – 1/p^s) in terms of the complementary arrangement of the two expressions

i.e. ∏ {1/(1 – 1/p^k) * (1 + 1/p^k)}

So for example, in the well-known case where s = 3,

∏ 1/(1 – 1/p³)  = ∏ {1/(1 – 1/p¹) * (1 + 1/p¹ + 1/p²)}, with no matching complementarity as between negative and positive terms in p.


And this complementarity (with respect to even terms) is also in evidence in the corresponding sum over (dimensional) natural numbers formulation provided through the Zeta 2 function.

So again the 1st term in the standard (Zeta 1) product formula expression where p (i.e. p¹) is 2 = 2/1.

And so the corresponding value given through the Zeta 2 expression,

= 1 + 1/p + 1/p² + 1/p³ +...   = 1 + 1/2 + 1/4 + 1/16 + ... = 2/1.

Now to express the corresponding (shadow) 1st term - related to the Zeta 1 - given as 1/(1 + 1/p¹) we simply alternate the signs in the Zeta 2 result to obtain

1  – 1/p¹ + 1/p²  – 1/p³ +...  = 1 – 1/2 + 1/4 – 1/8 + ... = 2/3.

And then by successively letting p = 3, 4, 5, ..... we can in a similar manner obtain all the other individual terms for the (shadow) product formulation of the Zeta 1 (where s = 1).

I earlier showed how to obtain the corresponding formulation (in the Zeta 2) for each product term (in the Zeta 1) where s = 2, through dropping each 2nd term in the original Zeta 2 expression (for s = 1).

Thus 4/3 is the 1st term in the product over primes, thereby  expressing, with s = 2, the value of 1/(1 – 1/p²) for the 1st prime, i.e. p = 2.

The corresponding Zeta 2 formulation is then obtained by dropping each 2nd term in the original formulation (for s = 1).

So 4/3 = 1 + 1/p² + 1/p⁴ + ... = 1 + 1/4 + 1/16 + ....


The corresponding (shadow) formulation of the Zeta 2 product expression, 1/(1 + 1/p²) = 4/5, is then given  in terms of the Zeta 2 as the alternating form of the previous expression for 1/(1 – 1/p²)

i.e. 4/5 = 1 – 1//p² + 1/p⁴ – ... = 1 – 1/4 + 1/16 – .... .


And once again we can obtain all the other individual terms in the (shadow) product expression for s = 2 by letting p = 3, 4, 5, ....

And just one more example to illustrate, where s = 3, we now all ordinal terms divisible by either 2 or 3!

So the 1st term in the (standard) product expression, where s = 3  = 8/7 (p = 2).

The corresponding Zeta 2 formulation is now given as

8/7  = 1 + 1/p³ + 1/p⁶ + ...    = 1 + 1/8  + 1/64 + ...

Then the 1st term in the corresponding (shadow) product expression represents the alternating version of the above series i.e.

8/9 = 1 – 1/p³ + 1/p⁶ – ...  = 1 – 1/8 + 1/64 – ...

Thus we now shown an important (positive/negative) complementarity governing terms, both with respect to the product over primes expression of the Zeta 1 and the corresponding sum over (dimensional) natural numbers of the Zeta 2.

And once again, this points directly to the inherent dynamic interactive nature of both Zeta 1 and Zeta 2 functions, which at a more fundamental level, illustrates the dynamic interactive nature of the number system itself with respect to the two-way relationship as between the primes and natural numbers.