Large values of Dirichlet polynomials
A limit theorem in the space of continuous functions for the Dirichlet polynomialwhere denote the coefficients of the Dirichlet series expansion of the function in the half-plane
The logarithmic derivative of the Γ-function, namely the ψ-function, has numerous applications. We define analogous functions in a four dimensional space. We cut lattices and obtain Clifford-valued functions. These functions are holomorphic cliffordian and have similar properties as the ψ-function. These new functions show links between well-known constants: the Eurler gamma constant and some generalisations, ζR(2), ζR(3). We get also the Riemann zeta function and the Epstein zeta functions.
We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments , where is a non-negative integer and a rational number. In particular, these lower bounds are of the expected order of magnitude for .
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...