MACHINERY OF ROSSER-SCHOENFELD METHOD FOR EXPLICIT APPROXIMATING OF PRIMES

We introduce the machinery of a method due to Rosser and Schoenfeld for approximating Chebychev functions in primenumberology, which ends to explicit approximation of primes. The story backs to the Riemann’s work on prime numbers. He guessed an explicit formula between the Chebychev function ψ (x) =S p^m≤ x log p and nontrivial zeros r =b +ig of the Riemann zeta function, defined for Re(s) > 1 by z (s) =S1/n^s and extended by meromorphic continuation to the complex plan. The connection (which is known as Riemann’s explicit formula) includes some elementary functions and the strange term limS|g|≤T x^r/r, and aim of Rosser-Schoenfeld method is approximating this summation. The procedure of method, needs some numerical and approximate data about non-trivial zeros of Riemann zeta function, which force us studying and applying zero free regions of this function. Also, it requires using complex analytic methods and theory of special functions.

کلیدواژه ها:

The Riemann zeta function ، prime numbers ، explicit approximation