On the Laurent Coefficients of Certain Dirichlet Series
The estimations of lower order in terms of the sequences and for an entire Dirichlet series , have been obtained, namely :One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).
For entire functions defined by absolutely convergent Dirichlet series, a theorem on their mean values is established which include the results of Kamthan, Juneja and Awasthi.
We study the supremum of some random Dirichlet polynomials , where (εₙ) is a sequence of independent Rademacher random variables, the weights (dₙ) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials , , P⁺(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for the supremum expectation that extend the optimal estimate of Halász-Queffélec, . The proofs are entirely based on methods of stochastic processes, in particular the metric...