I am posting this proposal on behalf of Dinesh Thakur.

Let be the ring of polynomials over the finite field of two elements, and let

be the set of irreducible polynomials in this ring. Then infinite series such as

and

can be expanded as formal infinite power series in the variable .

It was numerically observed in http://arxiv.org/abs/1512.02685 that one appears to have the remarkable cancellation

and

For instance, one has

and all other terms in are of order or higher, so this shows that has -valuation at least 3. Similarly, if one expands the first sum for all primes of degree (in ) up to 37, one obtains (the calculation took about a month on one computer), implying that the -valuation of the infinite sum is at least 38; in fact a bit of theory can improve this to 42. (But we do not know whether this 42 is the answer to everything!).

For the second sum, calculation for degrees up to 28 shows that the difference between the two sides has -valuation at least 88.

The polymath proposal is to investigate this phenomenon further (perhaps by more extensive numerical calculations) and supply a theoretical explanation for it.

Background links:

- The paper http://arxiv.org/abs/1512.02685 where these (and many more guesses of this type) are given with some background on zeta deformation etc, and
- http://www.math.rochester.edu/people/faculty/dthakur2/primesymmetryrev.pdf where the updated version is and will be maintained.

Below the fold is some more technical information regarding the above calculations.