On the Generalizations of Universality Theorem for L-Functions of Elliptic Curves

2017 ◽  
Vol 2 (44) ◽  
pp. 102-104
Author(s):  
Virginija Garbaliauskienė
Keyword(s):  
Positive Integer ◽  
Weak Convergence ◽  
Elliptic Curves ◽  
Limit Theorems ◽  
Probability Measures ◽  
Universality Theorem ◽  
Convergence Of Probability Measures ◽  
Derivatives Of

In the paper, the continuous type’s universality theorem for L-functions of elliptic curves is discussed and its generalizations in three directions – for positive integer powers and derivatives of L-functions of elliptic curves as well as the weighted universality theorem of L-functions of elliptic curves – are given. The proofs of the universality are based on limit theorems in the sense of weak convergence of probability measures in functional spaces.

A Weighted Universality Theorem for L-Functions of Elliptic Curves

2018 ◽  
Vol 48 (2) ◽  
pp. 18-21
Author(s):  
Antanas Garbaliauskas ◽  
Virginija Garbaliauskienė
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Elliptic Curves ◽  
Analytic Functions ◽  
Rational Numbers ◽  
Probability Measures ◽  
Space Of Analytic Functions ◽  
Continuous Type ◽  
Universality Theorem ◽  
Convergence Of Probability Measures

In the paper, a short survey on universality results for L-functions of elliptic curves over the field of rational numbers is given and weighted universality theorem is proven. All stated universality theorems are of continuous type. The proof of the universality for L-functions of elliptic curves is based on a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions.

A Discrete Limit Theorem for L-Functions of Elliptic Curves

2018 ◽  
Vol 48 (2) ◽  
pp. 27-29
Author(s):  
Virginija Garbaliauskienė ◽  
Antanas Garbaliauskas
Keyword(s):  
Limit Theorem ◽  
Probability Measure ◽  
Weak Convergence ◽  
Real Number ◽  
Elliptic Curves ◽  
Irrational Number ◽  
Probability Measures ◽  
Convergence Of Probability Measures ◽  
Main Statement

In the paper, we prove the discrete limit theorem in the sense of the weak convergence of probability measures in the space of analytic on DV = {s ∈ C : 1 < σ < 3/2, |t| <  V} functions for L-functions of elliptic curves LE(s). The main statement of the paper is as follows. Let h > 0 be a fixed real number such that exp {2πk/h} is an irrational number for all k∈Z\{0}. Then the probability measure μN(LE(s + imh)∈A), A ∈ B(H(DV)), converges weakly to the measure PLE as N→∞.

JOINT WEIGHTED LIMIT THEOREMS FOR GENERAL DIRICHLET SERIES

2011 ◽  
Vol 16 (1) ◽  
pp. 39-51
Author(s):  
Jonas Genys ◽  
Antanas Laurinčikas
Keyword(s):  
Weak Convergence ◽  
Dirichlet Series ◽  
Limit Theorems ◽  
Random Element ◽  
Probability Measures ◽  
Limit Measure ◽  
Additional Hypothesis ◽  
General Dirichlet Series ◽  
Convergence Of Probability Measures ◽  
Weighted Limit

In the paper,two joint weighted limit theorems in the sense of weak convergence of probability measures on the complex plane for general Dirichlet series are obtained. The first of them gives only the existence of the limit measure, while in the second theorem,under some additional hypothesis on the weight function, the explicit form of the limit measure is presented. Namely, the limit measure coincides with the distribution of some random element related to considered Dirichlet series.

scholarly journals Value-distribution of twisted L-functions of normalized cusp forms

2010 ◽  
Vol 51 ◽  
Author(s):  
Alesia Kolupayeva
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Complex Plane ◽  
Dirichlet Character ◽  
Value Distribution ◽  
Cusp Forms ◽  
Probability Measures ◽  
Convergence Of Probability Measures

A limit theorem in the sense of weak convergence of probability measures on the complex plane for twisted with Dirichlet character L-functions of holomorphic normalized Hecke eigen cusp forms with an increasing modulus of the character is proved.

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