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

2012 ◽  
Vol 08 (07) ◽  
pp. 1725-1740 ◽  
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].

A note on the mean value of L(1, χ) in the hyperelliptic ensemble

2014 ◽  
Vol 10 (04) ◽  
pp. 859-874 ◽  
Author(s):  
Hwanyup Jung
Keyword(s):  
Asymptotic Formula ◽  
Class Number ◽  
Mean Value ◽  
The Mean

In this paper, we establish an asymptotic formula for ∑D∈ℋ2g+2 L(1, χD) as g → ∞ (fixed odd q), where ℋ2g+2 is the subset of 𝔽q[T] consisting of monic square-free polynomials of degree 2g + 2. As an application, we obtain a formula for the average of the class number times the regulator of the associated rings [Formula: see text] when D is taken over ℋ2g+2 as g → ∞.

scholarly journals On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1254
Author(s):  
Xue Han ◽  
Xiaofei Yan ◽  
Deyu Zhang
Keyword(s):  
Asymptotic Formula ◽  
Fourier Coefficient ◽  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Mean Value ◽  
Symmetric Square ◽  
The Mean ◽  
Almost All

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

scholarly journals Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

2021 ◽  
Vol 56 (1) ◽  
pp. 79-94
Author(s):  
Nikola Lelas ◽  
Keyword(s):  
Rational Function ◽  
Finite Fields ◽  
Function Fields ◽  
Mean Value ◽  
Multiplicative Functions ◽  
Irreducible Polynomials ◽  
Liouville Function ◽  
The Mean

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

An approximate functional equation for Dirichlet L -functions

1965 ◽  
Vol 284 (1397) ◽  
pp. 224-236 ◽  
Keyword(s):  
Functional Equation ◽  
Asymptotic Formula ◽  
Royal Society ◽  
Approximate Functional Equation

A new asymptotic formula is derived for the computation of Dirichlet L -functions, L ( s , X ), where s = σ + i t . The formula is applicable for large values of t and it has been used on the Mercury computer at Manchester University to calculate the zeros of the L -functions with moduli 3 and 4 on the line Rs = ½. The results have been placed in the Royal Society Depository for Unpublished Mathematical Tables (no. 83).

scholarly journals The hybrid mean value of Dedekind sums and two-term exponential sums

2016 ◽  
Vol 14 (1) ◽  
pp. 436-442
Author(s):  
Chang Leran ◽  
Li Xiaoxue
Keyword(s):  
Asymptotic Formula ◽  
Exponential Sums ◽  
Mean Value ◽  
Gauss Sums ◽  
Mean Value Theorem ◽  
Dedekind Sums ◽  
Hybrid Mean Value ◽  
The Mean ◽  
Interesting Identity

AbstractIn this paper, we use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the two-term exponential sums, and give an interesting identity and asymptotic formula for it.

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