  We introduce a new set of prime numbers functions including an exact
Generating Function and a Discriminating Function of Prime Numbers neither
based on prime number tables nor on algorithms. Instead these functions are
defined in terms of ordinary elementary functions, therefore having the
advantage of being analytic and readily calculable. Also presented are four
applications of our new Prime Numbers Generating Function, namely: obtaining a
new analytic formula for counting prime numbers, obtaining an approximant to
Euler product function, obtaining an approximant to Riemann Zeta (sigma,tau)
function based on our primes discriminating function, an accurate approximant
to the Chebyshev function of second class in terms of our primes generating
function, and the application of this approximant in sharp estimates related to
the validity of the Riemann Hypothesis. We also apply the variational calculus
of classical mechanics to obtain the non-trivial roots of Riemann zeta function
in the complex plane, in an original and novel approach. A variational test
function based on the modulus squared of Riemann function is defined, and then
Hamilton Principle is applied to analytically obtain the non-trivial roots of
Riemann zeta in a completely original way. We optimize our analytical procedure
by defining a more general test function that depends explicitly on the
abscissa variable sigma, and present a procedure to find non-trivial roots
along the critical line, as demanded by Riemann Hypothesis, thus confirming it.
Our method even allows us to define a function, that behaves analogous to the
Riemann zeta function, and that even admits a functional type equation.
