scholarly journals SPECIAL VALUES OF SHIFTED CONVOLUTION DIRICHLET SERIES

Mathematika ◽  
2015 ◽  
Vol 62 (1) ◽  
pp. 47-66 ◽  
Author(s):  
Michael H. Mertens ◽  
Ken Ono
Keyword(s):  
Dirichlet Series ◽  
Special Values

scholarly journals Approximation by special values of Dirichlet series

2019 ◽  
Vol 148 (1) ◽  
pp. 83-93
Author(s):  
Şermin Çam Çelik ◽  
Haydar Göral
Keyword(s):  
Dirichlet Series ◽  
Special Values

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.

Rankin–Cohen brackets on Jacobi forms of several variables and special values of certain Dirichlet series

2019 ◽  
Vol 15 (05) ◽  
pp. 925-933
Author(s):  
Abhash Kumar Jha ◽  
Brundaban Sahu
Keyword(s):  
Dirichlet Series ◽  
Scalar Product ◽  
Fourier Coefficients ◽  
Differential Operators ◽  
Cusp Forms ◽  
Jacobi Forms ◽  
Linear Map ◽  
Special Values ◽  
Several Variables

We construct certain Jacobi cusp forms of several variables by computing the adjoint of linear map constructed using Rankin–Cohen-type differential operators with respect to the Petersson scalar product. We express the Fourier coefficients of the Jacobi cusp forms constructed, in terms of special values of the shifted convolution of Dirichlet series of Rankin–Selberg type. This is a generalization of an earlier work of the authors on Jacobi forms to the case of Jacobi forms of several variables.

ON A FORMULA FOR THE REGULARIZED DETERMINANT OF ZETA FUNCTIONS WITH APPLICATION TO SOME DIRICHLET SERIES

2020 ◽  
Vol 71 (3) ◽  
pp. 843-865
Author(s):  
Mounir Hajli
Keyword(s):  
Dirichlet Series ◽  
Large Class ◽  
Zeta Functions ◽  
Special Values

Abstract In this paper, we study a large class of zeta functions. We evaluate explicitly the special values of these zeta functions and the associated derivatives at $0$. As an application, we recover several results on the zeta functions defined by two polynomials already obtained in the literature.

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