Cauchy Means of Dirichlet polynomials
  We study Cauchy means of Dirichlet polynomials $$\int_\R \Big|\sum_{n=1}^N \frac{1}{ n^{\s+ ist}} \Big|^{2q}   \frac{\dd t}{\pi( t^2+1)}.$$   These integrals were investigated when $q=1,\s= 1, s=1/2 $ by Wilf, using integral operator theory and Widom's eigenvalue estimates.   We show the optimality of some upper bounds obtained by Wilf. We also obtain new estimates for the case $q\ge 1$, $\s\ge 0$ and $s>0$. We complete Wilf's approach by relating it with other approaches (having notably connection with Brownian motion), allowing simple proofs, and also prove new results. 
Michel Weber
math.NT math.CA