1.2. Dirichlet. A sequence of numbers a1, a2, . . . gives rise to a Dirichlet series f(s) = ∑ n≥1 ann −s. This is but a method of encoding the infinitely many numbers (an) into a single mathematical object f . If the numbers an grow at most polynomially, then f will actually be an analytic function on a half-plane { (s) > λ}; and the abscissa of convergence λ is related to the growth of the seq...

In this article, we obtain the analytic continuation of multiple shifted Lucas zeta function, L-function associated with Dirichlet characters and additive characters. We then compute a list possible singularities residues these functions at poles. Further, show rationality L-functions quadratic negative integer arguments.

We prove that for any real θ there are infinitely many values of s = σ + it with σ → 1+ and t→ +∞ such that Re {exp(iθ) logL(s, χ)} ≥ log log log log t log log log log t +O(1). The proof relies on an effective version of Kronecker’s approximation theorem. 1. Extremal values Extremal values of the Riemann zeta-function in the half-plane of absolute convergence were first studied by H. Bohr and L...

We show that if f(z) = ∑ anz n is a holomorphic function in the Dirichlet space of the unit disk, then almost all of its randomizations ∑ ±anz are multipliers of that space. This parallels a known result for lacunary power series, which also has a version for smoothness classes: every lacunary Dirichlet series lies in the Lipschitz class Lip1/2 of functions obeying a Lipschitz condition with ex...

In this work we propose a model-based clustering method for time series. The model uses an almost surely discrete Bayesian nonparametric prior to induce clustering of the series. Specifically we propose a general Poisson-Dirichlet process mixture model, which includes the Dirichlet process mixture model as particular case. The model accounts for typical features present in a time series like tr...