Prime numbers, Goldbach's conjecture, De Polignac's conjecture, Legendre's conjecture, Landau's conjecture, Mersenne's conjecture, and the Fermat number conjecture

Inspired by the article [3], the introduction of the function $\hat{\circledS}$ whose integer zeros are the prime numbers, will allow me to demonstrate analytically the Goldbach's conjecture [6], the De Polignac's Conjecture [7], the Legendre's conjecture [9], Landau's conjecture [10], Mersenne's conjecture [11] , and the Fermat number conjecture [12]