Cauchy means of Dirichlet polynomials

We study Cauchy means of Dirichlet polynomials ∫R|∑n=1N1nσ+ist|2qdtπ(t2+1). These integrals were investigated when q=1,σ=1,s=1/2q=1,σ=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≥1q≥1, σ≥0σ≥0 and s>0s>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.