On the Supremum of Random Dirichlet Polynomials Mikhail Lifshits and Michel Weber We study of some random Dirichlet polynomials P the supremum −σ−it DN (t) = N ε d n , where (εn ) is a sequence of independent n=2 n n Rademacher random variables, the weights (dn ) are multiplicative andP 0 ≤ σ < 1/2. The particular attention is given to the polynomials n∈Eτ εn n−σ−it , Eτ = {2 ≤ n ≤ N : P + (n) ≤ pτ }, P + (n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for supremum expectation that extend the optimal estimate of Hal´asz-Qu´effelec ¯ ¯ N ¯X ¯ N 1−σ ¯ −σ−it ¯ E sup ¯ . εn n ¯≈ ¯ log N t∈R ¯ n=2 Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method.


On the Supremum of Random Dirichlet Polynomials ...

On the Supremum of Random Dirichlet Polynomials. Mikhail Lifshits and Michel Weber. We study the supremum of some random Dirichlet polynomials. DN (t) =.

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