Abstract
Several aspects connecting analytic number theory and the Riemann zeta-function are studied and expanded. These include:
1. explicit formulae relating the Möbius function to the non-trivial zeros of the zeta
function;
2. generalized results on sums of Ramanujan sums;
3. new results on the combinations of Riemann $\Xi$-functions on bounded vertical shifts and their zeros on the critical line;
4. a generalization of moment integrals involving the Riemann $\Xi$-function;
5. asymptotics for the mean square of the product of the Riemann $\xi$-function and new Dirichlet polynomials;
6. zeta regularization on tori and a new proof of the Chowla-Selberg formula.
Robles, Nicolas