I proved in my article that Euclide's theorem about the infinity of primes contradict the asymptotic numbers of squarefree numbers and of primes. Do you have any ideas why this contradiction happened? Here is the link:
https://www.researchgate.net/publication/391942969_Euclid's_Theorem_against_the_asymptotic_numbers_of_squarefree_numbers_and_primes
This actually could be only that there's no dependence between them, but alone contradiction isn't sufficient.
Actually, there is no function that gives the value of square free forms for primes.
The ratio of odd squarefree numbers is 4/(pi^2) whereas the ratio of even squarefree numbers is 2/(pi^2).
The proof could be done if that would be shown using for example probabilistic method whether this is a contradiction or not.
In any case, this is a possible solution if we could use proof by induction where each odd squarefree number is at least larger as even one.
Only this contradiction between the Euclide's theorem and the asymptotic number of square-free numbers can allow the use of my upper bound of square-free numbers since the exact asymptotic number of square-free numbers should be revised.
I believe that the Basel problem is not solved yet. What do you think ?
Yes, indeed, there are more shades in mathematics than in any other science.
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