The Riemann Magneton of the Primes

Abstract

We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov. We define an integral m (r) involving the Zeta function in the complex variable s = r + it and find a particurarly interesting expression for m(r) which is rigorous at least in some range of r. In such a range we find that there are two discontinuities of the derivative m'(r) at r = 1 and r = 0, which we calculate exactly. The jump at r = 1 is given by 4*Pi, while that at r = 0 is given by (-4 + gamma + 3*log(2) + Pi/2)*Pi. The validity of the expression for m(r) up to r = 1/2 is equivalent to the truth of the Riemann Hypothesis (RH). Assuming RH the expression for m(r) gives m = 0 at r = 1/2 and the slope m'(r) = Pi*(1 + gamma) = 4.95 at r = 1/2 (where gamma = 0.577215... is the Euler constant). As a consequence, if the expression for m(r) can be continued up to r = 1/2, then if we interpret m(r) as the effective potential of a charged filament, in the presence of the unit charges located at all the zeros (trivial and non trivial zeros) and at the pole in one of the Riemann Zeta function then there is a jump of the electric field E(r) = m'(r) at r = 1/2 given by 2*Pi times the first Li's coefficient lambda_1 = [1+gamma/2-(1/2)ln(4*Pi)] = 0.0230957. We then construct a potential well (a symmetric function around x = r+1/2 = 1) which is exact if the RH is true. Independently of the RH, by looking at the behaviour of the convergent Taylor expansion of m(r) at r = 1-, the value m(r = 1/2+) as well as the first Li's coefficient may be evaluated using the Euler product formula. We give in this way further evidence for the possible truth of the Riemann Hypothesis.