Musings on the Riemann Hypothesis

Abstract

We present a few ideas on the Riemann Hypothesis based on properties of analytic functions in the complex plane. In particular, we focus on the real and imaginary parts of the Riemann xi ($ξ$) function whose zeros coincide with those of the zeta ($ζ$) function within the critical strip. We discuss the forms of the zero contour lines of the two conjugate harmonic functions (the real and imaginary parts of xi) and consider where their intersections could conceivably occur. Those intersections would be the roots of both $ξ$ and $ζ$ functions. The question of whether a zero could occur away from the critical line becomes equivalent to whether the solutions of a pair of Laplace's equations with well-defined boundary conditions in some semi-infinite strip can possess zero contour lines that intersect within that strip.