Modulus representation of the Riemann $\hat{I}¾$ function and polynomial inequalities equivalent to the Riemann hypothesis

Abstract

We use the Jacobi theta function to give a representation of the modulus of the Riemann $\hat{I}¾$ function. Based on this modulus representation, we show that the Riemann hypothesis is equivalent to the validity of a family of polynomial inequalities. We also present some preliminary results on the polynomial inequalities.