Relation between primes and nontrivial zeros in the Riemann hypothesis; Legendre polynomials, modified zeta function and Schrödinger equation

Abstract

We study the dependence between prime numbers and the real and imaginary parts of the nontrivial zeros of the Riemann zeta function. The Legendre polynomials and the partial derivatives of the Riemann zeta function are used to investigate the above dependence along with the Riemann hypothesis with physical interpretations. A modified zeta function with finite terms is defined as a new implement for the study of the zeta function and its zeros.