A Hidden Symmetry Related to the Riemann Hypothesis with the Primes into the Critical Strip

In this note concerning integrals involving the logarithm of the Riemann Zeta function, we extend some treatments given in previous pioneering works on the subject and introduce a more general set of Lorentz measures. We rst obtain two new equivalent formulations of the Riemann Hypothesis (RH). Then with a special choice of the measure we formulate the RH as a hidden symmetry , a global symmetry which connects the region outside the critical strip with that inside the critical strip. The Zeta function with all the primes appears as argument of the Zeta function in the critical strip. We then illustrate the treatment by a simple numerical experiment. The representation we obtain go a little more in the direction to believe that RH may eventually be true. 1. Some integrals involving the Zeta function We start with some integrals involving the absolute values of the logarithm of the Zeta function. As far as we know, the rst work in this direction is due to Wang, who discovered a RH criterium involving these integrals [1]. More recent pioneering works are due to Volchkov [2] who found an integral relation on the complex plane with two variables equivalent to the Riemann Hypothesis (RH). Later Balazard, Saias and Yor [3], established another equivalence to the RH by an integral involving only one variable i.e. by integration on the critical line. In a subsequent treatment by one of us the analytical computations were extended to every line perpendicular to the x axis to obtain an equivalence to the RH involving explicitly R(s) = ρ, with the appearance of a shift along the real axis of exactly 12 [4]. In the present note we rst extend some of the above mentioned treatments by introducing a more general Lorentz measure which we are free to normalize in order to obtain more simple formulas i.e.: