scholarly journals Analytic Continuation of Random Dirichlet Series

10.4213/spm44 ◽  
2013 ◽  
Vol 17 ◽  
pp. 76-81
Author(s):  
Gautami Bhowmik ◽  
Gautami Bhowmik ◽  
К Матсумото ◽  
Kohji Matsumoto
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Random Dirichlet Series

Analytic continuation of a Dirichlet series

1987 ◽  
Vol 42 (5) ◽  
pp. 845-848
Author(s):  
S. A. Zakharov
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series

On Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

Deficient functions of random dirichlet series

2007 ◽  
Vol 27 (2) ◽  
pp. 291-296
Author(s):  
Junying Zhou ◽  
Daochun Sun
Keyword(s):  
Dirichlet Series ◽  
Random Dirichlet Series

scholarly journals SPECIAL VALUES OF DIRICHLET SERIES AND ZETA INTEGRALS

2012 ◽  
Vol 08 (03) ◽  
pp. 697-714 ◽  
Author(s):  
EDUARDO FRIEDMAN ◽  
ALDO PEREIRA
Keyword(s):  
Positive Integer ◽  
Analytic Continuation ◽  
Dirichlet Series ◽  
Direct Calculation ◽  
Simple Relation ◽  
Product Rule ◽  
Special Values ◽  
Special Value

For f and g polynomials in p variables, we relate the special value at a non-positive integer s = -N, obtained by analytic continuation of the Dirichlet series [Formula: see text], to special values of zeta integrals Z(s;f,g) = ∫x∊[0, ∞)p g(x)f(x)-s dx ( Re (s) ≫ 0). We prove a simple relation between ζ(-N;f,g) and Z(-N;fa, ga), where for a ∈ ℂp, fa(x) is the shifted polynomial fa(x) = f(a + x). By direct calculation we prove the product rule for zeta integrals at s = 0, degree (fh) ⋅ Z(0;fh, g) = degree (f) ⋅ Z(0;f, g) + degree (h) ⋅ Z(0;h, g), and deduce the corresponding rule for Dirichlet series at s = 0, degree (fh) ⋅ ζ(0;fh, g) = degree (f) ⋅ ζ(0;f, g)+ degree (h)⋅ζ(0;h, g). This last formula generalizes work of Shintani and Chen–Eie.

scholarly journals The order of certain Dirichlet series

1971 ◽  
Vol 12 (4) ◽  
pp. 441-443
Author(s):  
Chung-Ming An
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Homogeneous Part ◽  
Image Position

This paper is a continuation of [1]1 We shall use the same notations as those in [1]. Let F(X) ∈ R[X], X = (X1, …, Xn), be a polynomial of degree d > 0 and h(x) ∈ SP(Rn), i.e. h(x) is the sum of a polynomial and a Schwartz funtion. We shall consider Dirichlet series of the type where NF = {x∈Rn: F(x) = 0}. We proved, in [1], that Z(h, F, s) is regular for σ > (n+p)/d and possesses the analytic continuation to the whole s-plane when Fd(x) (the highest homogeneous part of F(X)) ≠ 0 for x ≠ 0. In this paper, we shall say the following.

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