# Twisted second moments and explicit formulae of the Riemann zeta-function

Robles, Nicolas. Twisted second moments and explicit formulae of the Riemann zeta-function. 2015, University of Zurich, Faculty of Science.

## 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.

## 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.

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