Thinking like a Zeta Function

The thing above.Give me a sequence of elements having commutative property.  If I grab an infinity of these, what is the measure of them happening S times, counting duplicates so this does not converge to one. The s has to be greater than one one.  What is s contains imaginary sequences, things happening in hologram space? We let s have an inferred line of symmetry. Does it contain a measure of the duplicaets?



As a finite precision estimator
Consider the related problem, we want to take some finite integer less and N and encode it to its most likely nearest prime, to some fixed precision.  So, in the complement, we envision a generator of likely primes, driven by a uniform counting sequence. We have switched from encoding primes to decoding them.

If I change my finite bounds to a window, and move that window up and down the number line, then smaller primes get bundled, we get the Walmart checkout manager setting the 'primes per basket'.  He has allocated primes to keep the fixed size baskets optimal.  Then, we ask a different question, does the requantization and compression obey the same prime commutative properties? That is, does moving the window simply reuse a set of lower level primes, making the system analytic? Then we can bound the error in counting  prime patterns. The generator maintains form as it adiabatically moves across the number line. No it does not, if we move  fixed segment out long enough, it will encompass only one prime, the generator has rank zero. There is wasted information, relative to prime division, as a fixed index window moves right.

Let us fix the precision of our probabilistic prime generator, but let the window grow and shift right. Then as it grow right and primes becomes scarce eventually the largest prime splits the group, and rank increases in the generator, the short end of the window must move wight and bundle one or more of the small primes, in a compression.  Now, the question, just as difficult, is how does that window grow as the N grows?

Check all numbers from one to N, and as N expands past another prime, the rank of our generator goes up. But the nodes not necessarily lined at even splits, the splits are unknown, the the problem becomes how and what dose adiabatic  transform of generating mean. Yes primes get sparse, but how can we bound the process? That is the great question.


Try a solution
We have a fixed precision generator of nearest primes over an expanding bound window on the integer set. In the compression induced, there is a space left, in the integer line, unattended uncertainty, the probability that the nearest prime is too far left to be noticed. But we have seen that probability, from some earlier in the sequence of moving right. We have a recursion, we can find zeros.  This is what we mean in fitting the Riemann to a quantizing system.

My little approaches illustrates an important point. N, the current size of an expanding window is the Avogadro under the select precision of my generator. It expands, it is variable.  Zeta is a general problem, sphere packers are a finite set of solutions of the thing, known in many cases.

So, tell me your uncertainty, and I can give you an Avogadro number that reserves just enough prime space for uncertainty. Because I know the rank, I know the window moves right and leaves an uncertainty barrier to the left; thus I have a recursion,show an asymptote for any rank generator . No! Send me a banana. No? Why not? My analytic extension needs a polynomial set that expands with the same algebra. There is another half of the proof, and I have no more lifetimes.

Models of information compression and lossy expansion are just variations of Taylor rules, locally approximating the rules of Newton. But we will have a divergence problem, proofs are hard.