[VA] SRC #012e - Then and Now: Roots

[J-F Garnier]

This has been a difficult problem for me (at different levels) that seems so simple to solve when reading Valentin's solution.
As I often noticed, carefully reading the problem statement is already a part of the solution. And indeed, contrary to most of Valentin's problems, using the Web resources was not explicitly prohibited here. I missed that aspect at the beginning.
This was the clue that we really needed: don't limit our attempts to our own (limited) resources, think wider.

(02-26-2023 12:34 AM)Valentin Albillo Wrote:  Actually, I think it's indeed a problem with a most amazing, awesome result, where a simple-looking function in its canonical (and Gram series, too) form, with no ad-hoc constants or contrivances and whose graph is a dull logarithmic-like curve for macroscopic arguments, suddenly turns out wiggly and crosses the X-axis by the tiniest amount when evaluated at "nanoscopic" arguments (0.00000...{14,000+ zeros}...0000018286...), completely invisible and utterly unexpected.

In my first investigations, I tried to evaluate the summation (the Gram series) without the zeta factors, considering that they quickly tend to 1 and should have little impact, so I may have a first estimation.
Unfortunately, that summation doesn't tend to zero anymore, actually not even to a finite limit when X tend to zero. So my attempt failed.
This was a major surprise to me, much more than the tiny roots: the limit to zero is due to these zeta terms. It may have a deep mathematic meaning but it's out of my understanding.

J-F