We will show how to combine the Fast Fourier Transform algorithm with the reflection formulae of the special functions involved in the computation of the values of
and
, where
runs over the Dirichlet characters modulo an odd prime number
. In this way, we will be able to reduce the memory requirements and to improve the computational cost of the whole procedure.
Several applications to number-theoretic problems will be mentioned, like the study of the distribution of the Euler-Kronecker constants for the cyclotomic field and its subfields, the behaviour of
, the study of the Kummer ratio for the first factor of the class number of the cyclotomic field and the ``Landau vs. Ramanujan`` problem for divisor sums and coefficients of cusp forms. Towards the end of the seminar we will tackle open problems both of theoretical and implementative nature.