arXiv subjects
Publications and source records attributed to Giovanni Lodone.
The concept of angular momentum is used to find new RH equivalence statements, and, generalize some known results from Riemann to Dirichlet primitive Xi functions
First idea is to compute a quantity like the angular momentum with respect to (0, 0), of an unitary mass of coordinates (<[Xi(s)], =[Xi(s)]) while =[s] is the time, and, <[s] = constant. If we impose that the derivative along <[s], at points <[s] = 1/2 is grater than zero, then, we find exactly a known RH equivalence statement about relative maxima and minima of Xi(1/2 + i=[s]) along critical line. After representing this fictitious angular momentum by Euler Product, and, using PNT as a tool, it can be proved that this positivity condition is granted everywhere at least for Xi(1/2 + i=[s]) 6 = 0. So, if the above equivalence is true, it is found that off-critical line zeros must be excluded for Z(s) function along all critical strip . Further analysis on Euler Product(Lemma 2) has evidenced others shorter ways to same objective. Besides the converging spectrum of prime numbers is highlighted as a by-product.
We try to apply a known equivalence, for RH about Riemann Z function, to Dirichlet L functions with primitive characters. The aim is to give a small contribution to the proof of the generalized version of Riemann Hypothesis (RH).
The usual Riemann-Siegel Z(t) is a real-valued function. We construct a complex function depending from t and from distance from critical line. It is linked to Riemann Xi(s) function by the same real scaling factor of the usual Riemann-Siegel Z(t) on critical line. Errors are not greater than the errors of Riemann-Siegel Z(t) on the critical line, while this result covers at least the whole critical strip.
An approximate formula for complex Riemann Xi function, previously developed, is used to refine Backlund's estimate of the number of zeros till a chosen imaginary coordinate