On a certain class of dirichlet series

For 0 1, let L(s,a) and L?(s,a) be the Dirichlet series L(s,a) = ∑ : cos (2πna)n^-s and L? [001](s,a) = ∑ sin (2πna)n^-s. We show that L(s,a) and L?(s,a[001]) have holomorphic extension in the whole complex plane. Values of L(s,a)andL?(s,a) at the negative integers are given. Moreover values of L?(s,a) at the intergers 0,2,4,... and values of L^?(s,a)at the integers 1,3,5,... are obtained. An exponential sums of certain recursion formulas are obtained by means of bernoulli numbers and Bernoulli polynomials     

Rationality of a certain formal power series attached to local densities of quadratic forms

We will prove the rationality of the power series attached to local densities defined by the author     

Analytic continuation of a class of dirichlet series

We consider Dirichlet series of the type Σ(logk)ⁿ(k)(logk)?^-s. We prove the existence of an analytic continuation to the cut plane and give exact information about the singularity. We use this to generalize results, which occur in Ramanujan’s second notebook.     

Exact estimates for integrals related to dirichlet series

First, we consider integrals of the form $$\int {a(x)m^{ - x} dx for} m = 2,3,...$$ over the unit interval (0, 1) or the interval (1, ∞) or the half-line (0, ∞), wherea(x)≥0 and is integrable on the interval in question. These integrals are related to the Dirichlet series $$\sum\limits_{m = 2}^\infty {a_m m^{ - x} for x > 1} ,$$ , where the numbersa _m ≥0. We survey certain known results in a new formulation in order to reveal the difference in behavior between the functions which are integrable on either (0, 1) or (1, ∞). Their proofs can be read out from the existing literature. Second, we extend these results from single to double related integrals, while making distinction among the functionsa(x, y) which are integrable on either (0, 1)² or (0, 1)×(1, ∞) or (1, ∞)×(0, 1) or (1, ∞)². The case wherea(x, y) is integrable on (0, ∞)² is also included.     

DEFICIENT FUNCTIONS OF RANDOM DIRICHLET SERIES

In this article, the uniqueness theorem of Dirichlet series is proved. Then the random Dirichlet series in the right half plane is studied, and the result that the random Dirichlet series of finite order has almost surely(a.s.) no deficient functions is proved.     

On certain relations between the maximum modulus and the maximal term of an entire dirichlet series

For an entire Dirichlet series , sufficient conditions on the exponents are established such that the following relations hold outside a set of finite measure asx→+∞:

Julia lines of general random dirichlet series

In this paper, we consider a random entire function f(s, ω) defined by a random Dirichlet series $\sum\nolimits_{n = 1}^\infty {{X_n}(w\omega ){e^{ - {\lambda _n}s}}} $ where X _n are independent and complex valued variables, 0 ? λ _n ↗ +∞. We prove that under natural conditions, for some random entire functions of order (R) zero f(s, ω) almost surely every horizontal line is a Julia line without an exceptional value. The result improve a theorem of J.R.Yu: Julia lines of random Dirichlet series. Bull. Sci. Math. 128 (2004), 341–353, by relaxing condition on the distribution of X _n for such function f(s, ω) of order (R) zero, almost surely.