Generalized Bombieri–Lagarias’ Theorem and Generalized Li’s Criterion with its Arithmetic Interpretation
Generalized Bombieri–Lagarias’ Theorem and Generalized Li’s Criterion with its Arithmetic Interpretation
We show that Li’s criterion equivalent to the Riemann hypothesis, i.e., the statement that the sums $$ {k}_n={\Sigma}_{\uprho}\left(1-{\left(1-\frac{1}{\uprho}\right)}^n\right) $$ over zeros of the Riemann xi-function and the derivatives $$ \begin{array}{ccc}\hfill {\uplambda}_n\equiv \frac{1}{\left(n-1\right)!}\frac{d^n}{d{z}^n}{\left.\left({z}^{n-1} \ln \left(\upxi (z)\right)\right)\right|}_{z=1},\hfill & \hfill \mathrm{where}\hfill & \hfill n=1,2,3,\dots, \hfill \end{array} $$ are nonnegative if and only if …