Approximate functional equation and mean value formula for the derivatives of $L$-functions attached to cusp forms

AbstractIn this series of papers, we explore moments of derivatives of L-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials P in 𝔽q[T]. When the cardinality q of the ground field is fixed and the degree of P gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of L-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.

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

International Journal of Number Theory ◽

10.1142/s1793042108001730 ◽

2008 ◽

Vol 04 (05) ◽

pp. 747-756 ◽

Cited By ~ 2

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.

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The MacLaurin series for the GI/G/1 queue

Journal of Applied Probability ◽

10.2307/3214801 ◽

1992 ◽

Vol 29 (1) ◽

pp. 176-184 ◽

Cited By ~ 35

Author(s):

Wei-Bo Gong ◽

Jian-Qiang Hu

Keyword(s):

Service Time ◽

Mean Value ◽

Interarrival Time ◽

Service Time Distribution ◽

Maclaurin Series ◽

Recursive Procedure ◽

Derivatives Of ◽

System Time ◽

Time Density ◽

Mean Value Formula

We derive the MacLaurin series for the moments of the system time and the delay with respect to the parameters in the service time or interarrival time distributions in the GI/G/1 queue. The coefficients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The light traffic derivatives can be obtained from these series. For the M/G/1 queue, we are able to recover the formulas for the moments of the system time and the delay, including the Pollaczek–Khinchin mean-value formula.

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A NOTE ON THE MEAN VALUE OF L-FUNCTIONS IN FUNCTION FIELDS

International Journal of Number Theory ◽

10.1142/s1793042112500996 ◽

2012 ◽

Vol 08 (07) ◽

pp. 1725-1740 ◽

Cited By ~ 6

Author(s):

JULIO ANDRADE

Keyword(s):

Functional Equation ◽

Asymptotic Formula ◽

Finite Field ◽

Class Number ◽

Function Fields ◽

Hyperelliptic Curves ◽

Mean Value ◽

Approximate Functional Equation ◽

The Mean

An asymptotic formula for the sum ∑ L(1, χ) is established for a family of hyperelliptic curves of genus g over a fixed finite field 𝔽q as g → ∞ making use of the analog of the approximate functional equation for such L-functions. As a corollary, we obtain a formula for the average of the class number of the associated rings [Formula: see text].

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