## Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series

### Volume 84 / 1998

Acta Arithmetica 84 (1998), 155-192
DOI: 10.4064/aa-84-2-155-192

#### Abstract

We obtain formulas for computing mean values of Dirichlet polynomials that have more terms than the length of the integration range. These formulas allow one to compute the contribution of off-diagonal terms provided one knows the correlation functions for the coefficients of the Dirichlet polynomials. A smooth weight is used to control error terms, and this weight can in typical applications be removed from the final result. Similar results are obtained for the tails of Dirichlet series. Four examples of applications to the Riemann zeta-function are included.