The Landau's problems.II: Landau's problems solved

This is the second part of a work devoted to the four problems in Number Theory, called {\em Landau's problems}, listed by Edmund Landau at the 1912 International Congress of Mathematicians. These problems are the following. 1. Goldbach's conjecture. 2. Twin prime conjecture. 3. Legendre's conjecture. 4. Are there infinitely many primes $p$ such that $p-1$ is a perfect square ? In part I the proof of the Goldbach's conjecture has been already given. In this paper we show that by utilizing some algebraic topologic methods introduced in Part I, some Landau's problems can be proved too. Furthermore, for the above fourth Landau's problem a Euler-Riemann zeta function estimate is given and settled the problem negatively by evaluating the cardinality of the set of solutions of a suitable Diophantine equation of Ramanujan-Nagell-Lebesgue type.