Extension of the Titchmarsh representation of the Mertens function M(x)M(x), and numerical support of the Riemann hypothesis

A closed form representation of the Mertens function, without assuming simplicity of the non-trivial zeros of the zeta function, is proved and the Titchmarsh representation of the function is recovered as a special case. We exploit a representation of the Mertens function and provide numerical support of the Riemann hypothesis and the simplicity of the zeros of the zeta function.