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.

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.

A new asymptotic representation for ζ(½ + i t ) and quantum spectral determinants

1992 ◽  
Vol 437 (1899) ◽  
pp. 151-173 ◽  
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
Dirichlet Series ◽  
Asymptotic Representation ◽  
Critical Line ◽  
Error Function ◽  
Real Variable ◽  
Computational Effort ◽  
Selberg Zeta Function ◽  
Semiclassical Approximations

By analytic continuation of the Dirichlet series for the Riemann zeta function ζ(s) to the critical line s = ½ + i t ( t real), a family of exact representations, parametrized by a real variable K , is found for the real function Z ( t ) = ζ(½ + i t ) exp {iθ( t )}, where θ is real. The dominant contribution Z 0 ( t,K ) is a convergent sum over the integers n of the Dirichlet series, resembling the finite ‘main sum ’ of the Riemann-Siegel formula (RS) but with the sharp cut-off smoothed by an error function. The corrections Z 3 ( t,K ), Z 4 ( t,K )... are also convergent sums, whose principal terms involve integers close to the RS cut-off. For large K , Z 0 contains not only the main sum of RS but also its first correction. An estimate of high orders m ≫ 1 when K < t 1/6 shows that the corrections Z k have the ‘factorial/power ’ form familiar in divergent asymptotic expansions, the least term being of order exp { ─½ K 2 t }. Graphical and numerical exploration of the new representation shows that Z 0 is always better than the main sum of RS, providing an approximation that in our numerical illustrations is up to seven orders of magnitude more accurate with little more computational effort. The corrections Z 3 and Z 4 give further improvements, roughly comparable to adding RS corrections (but starting from the more accurate Z 0 ). The accuracy increases with K , as do the numbers of terms in the sums for each of the Z m . By regarding Planck’s constant h as a complex variable, the method for Z ( t ) can be applied directly to semiclassical approximations for spectral determinants ∆( E, h ) whose zeros E = E j ( h ) are the energies of stationary states in quantum mechanics. The result is an exact analytic continuation of the exponential of the semiclassical sum over periodic orbits given by the divergent Gutzwiller trace formula. A consequence is that our result yields an exact asymptotic representation of the Selberg zeta function on its critical line.

scholarly journals Distributions and analytic continuation of Dirichlet series

2004 ◽  
Vol 214 (1) ◽  
pp. 155-220 ◽  
Author(s):  
Stephen D Miller ◽  
Wilfried Schmid
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series
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