scholarly journals On the Error Term in Duke's Estimate for the Average Special Value of L-Functions

2005 ◽  
Vol 48 (4) ◽  
pp. 535-546 ◽  
Author(s):  
Jordan S. Ellenberg
Keyword(s):  
Error Term ◽  
Orthonormal Basis ◽  
Cusp Forms ◽  
Nonzero Constant ◽  
Weighted Averages ◽  
Special Values ◽  
Special Value

AbstractLet be an orthonormal basis for weight 2 cusp forms of level N. We show that various weighted averages of special values L( f ⭙ χ, 1) over f ∈ are equal to 4πc +O(N–1+∊), where c is an explicit nonzero constant. A previous result of Duke gives an error term of O(N–1/2 log N).

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 THE ERROR TERM IN THE APPROXIMATE FUNCTIONAL EQUATION FOR EXPONENTIAL SUMS RELATED TO CUSP FORMS

2008 ◽  
Vol 04 (05) ◽  
pp. 747-756 ◽  
Author(s):  
ANNE-MARIA ERNVALL-HYTÖNEN
Keyword(s):  
Functional Equation ◽  
Upper Bound ◽  
Error Term ◽  
Exponential Sums ◽  
Cusp Forms ◽  
Approximate Functional Equation ◽  
Holomorphic Cusp Forms

We give a proof for the approximate functional equation for exponential sums related to holomorphic cusp forms and derive an upper bound for the error term.

Special values of Koecher–Maass series of Siegel cusp forms

2001 ◽  
Vol 198 (2) ◽  
pp. 373-383 ◽  
Author(s):  
YoungJu Choie ◽  
Winfried Kohnen
Keyword(s):  
Cusp Forms ◽  
Special Values
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