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We introduce a new set of prime numbers functions including an exactGenerating Function and a Discriminating Function of Prime Numbers neitherbased on prime number tables nor on algorithms. Instead these functions aredefined in terms of ordinary elementary functions, therefore having theadvantage of being analytic and readily calculable. Also presented are fourapplications of our new Prime Numbers Generating Function, namely: obtaining anew analytic formula for counting prime numbers, obtaining an approximant toEuler product function, obtaining an approximant to Riemann Zeta (sigma,tau)function based on our primes discriminating function, an accurate approximantto the Chebyshev function of second class in terms of our primes generatingfunction, and the application of this approximant in sharp estimates related tothe validity of the Riemann Hypothesis. We also apply the variational calculusof classical mechanics to obtain the non-trivial roots of Riemann zeta functionin the complex plane, in an original and novel approach. A variational testfunction based on the modulus squared of Riemann function is defined, and thenHamilton Principle is applied to analytically obtain the non-trivial roots ofRiemann zeta in a completely original way. We optimize our analytical procedureby defining a more general test function that depends explicitly on theabscissa variable sigma, and present a procedure to find non-trivial rootsalong the critical line, as demanded by Riemann Hypothesis, thus confirming it.Our method even allows us to define a function, that behaves analogous to theRiemann zeta function, and that even admits a functional type equation.

New generating and counting Functions of prime numbers applied to  approximate Chebyschev 2nd class function and the least action principle  applied to find non-trivial roots of the Zeta function and to Riemann  Hypothesis