Generalisation of Result

In general terms with reference to the prototype Riemann zeta function, if we choose the primes (in the product over primes expression) using any infinite ordered sequence of integers, then a corresponding sum over the natural numbers expression can thereby be provided.

For example the (infinite) series of triangular numbers i.e. 1, 3, 6, 10, 15, …, provides one such ordered sequence (where we are always enabled to provide the next term in the sequence i.e. n(n + 1)/2, with n = 1, 2, 3, …

So if we now apply this ordering to the primes by choosing the 1^st, 3^rd, 6^th, 10^th 15^th … numbers, then we have 2, 5, 13, 29, 47, …

So the corresponding sum over the integers expression is thereby now based (including 1) on using these primes as sole factors.

Thus

1/1^s + 1/2^s +1/4^s + 1/5^s + 1/8^s + 1/10 ^s …  = 1/(1 – 1/2^s) * 1/(1 – 1/5^s) * 1/(1 – 1/13^s) * …

In fact the prototype Riemann zeta function itself corresponds on an ordering that is equivalent to the denominators of the same zeta function (for s = 1).

So if we now take an ordering based on the denominators of the function for s = 2, it will correspond to the sum of squares 1, 4, 9, 16, 25,…

Thus choosing our primes based on these ordinal rankings we have 2, 7, 23, 53, 97,…

Then the sum over integers expression matching the product over primes expression (using this selection) is based solely (including 1) on numbers based on the factors of such primes.

Thus,

1/1^s + 1/2^s +1/4^s + 1/7^s + 1/8^s + 1/14 ^s …  = 1/(1 – 1/2^s) * 1/(1 – 1/7^s) * 1/(1 – 1/23^s) * …

Of course as always we can then choose to omit any specific prime (or combination of primes) from the RHS expression with corresponding adjustments (based on omission of natural numbers where these are factors) on the LHS.

Therefore in choosing to omit 2 from the RHS (product over primes), we thereby choose to omit all numbers with 2 as a common factor in the LHS (sum over the integers) expression.   

So,

1/1^s + 1/7^s + 1/23^s + 1/49^s + 1/53^s + …  = 1/(1 – 1/7^s) * 1/(1 – 1/23^s) * 1/(1 – 1/53^s) * …

When in the above s = 4 the sum of listed terms for LHS expression = 1.000420367, while the corresponding sum of terms for RHS expression = 1.000420368 (which already shows an extremely close matching).