A 2-variable power series approach to the Riemann hypothesis
Abstract
We consider the power series in two complex variables By(fb)(x)=S(n=0)|.Anb xn y(n(n+1)/2)., where .(-1).n Anb are the non-zero coefficients of the Maclaurin series of the Riemann Xi function. The Riemann hypothesis is the assertion that all zeros of B1 (fb) are real. We prove that every zero of By (fb) is the inverse of a power series in y with real coefficients, which converges for |y|<0,2078.... We show the existence of a constant T, similar to the de Bruijn-Newman constant, satisfying : 0= y =T if and only if all zeros of By (fb) are real. We prove that 1/4 = T = 1 and that T=1 is equivalent to the Riemann hypothesis. We show that the Riemann hypothesis is also equivalent to what the discriminant of each Jensen polynomial of By (fb) does not vanish on the interval [1/4,1|[. We prove the Riemann hypothesis implies that the zeros of By (fb) are simple for 0<y<1, and we conjecture that the reciprocal implication is true.