On mean values of Dirichlet polynomials (Q1193273)

For fixed non-zero real numbers \(\alpha\) and \(\beta\) it is shown that \[ \int_ 0^ T|\sum_{N<n\leq 2N} a_ n n^{i\alpha t} |^ 2\;|\sum_{M<m\leq 2M} b_ m m^{i\beta t} |^ 2\;dt\ll(T+MN)MN\log 2MNT, \] for \(T,M,N\geq 1\) and arbitrary coefficients satisfying \(| a_ n|,| b_ m|\leq 1\). When \(\alpha=\beta\) this is an easy consequence of the usual mean-value estimate for a single Dirichlet polynomial, but the general case appears to require treatment from first principles. The proof depends on an estimate for the number of solutions of the inequality \[ \left|\left({n'\over n}\right)^ \alpha-\left({m'\over m}\right)^ \beta \right|\leq\Delta. \] The authors also discuss the possible extensions of their result to products of 3 or more Dirichlet polynomials, and give a partial result using mean-values of the Riemann zeta-function.