A test for the Riemann hypotesis

Abstract

We prove that the Riemann Hypothesis holds if and only if I = Z +∞ 1 ˘ Π(x) − Li(x) ¯2 x −2 dx < +∞ with I = J, where J is some definite, computable real number (1.266 < J < 1.273). This provides us with a numerical test for the Riemann Hypothesis. The main interest of our test lies in the fact that it can also supply a goal. Namely, having computed J(a) := R a 1 ˘ Π(x) − Li(x) ¯2 x −2 dx < J for a number of values of a = an, we can estimate a value a for which, within our precision, we will have J(a) ≈ J.