Statistics of Hecke eigenvalues for GL(𝑛)

2019 ◽  
Vol 31 (1) ◽  
pp. 167-185
Author(s):  
Yuk-Kam Lau ◽  
Ming Ho Ng ◽  
Yingnan Wang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Covariance Matrix ◽  
Central Limit ◽  
Cusp Forms ◽  
Statistical Independence ◽  
Two Dimensional ◽  
Hecke Operator ◽  
Hecke Eigenvalues ◽  
Compact Region

Abstract A two-dimensional central limit theorem for the eigenvalues of {\mathrm{GL}(n)} Hecke–Maass cusp forms is newly derived. The covariance matrix is diagonal and hence verifies the statistical independence between the real and imaginary parts of the eigenvalues. We also prove a central limit theorem for the number of weighted eigenvalues in a compact region of the complex plane, and evaluate some moments of eigenvalues for the Hecke operator {T_{p}} which reveal interesting interferences.

scholarly journals Statistical independence in mathematics–the key to a Gaussian law

2020 ◽  
Author(s):  
Gunther Leobacher ◽  
Joscha Prochno
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Historical Context ◽  
Lacunary Series ◽  
Statistical Independence ◽  
Binary Expansion ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
Kolmogorov Axioms

Abstract In this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the 1930s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. Among other things, we present the independence of the coefficients in a binary expansion together with a central limit theorem for the sum-of-digits function as well as the independence of divisibility by primes and the resulting, famous central limit theorem of Paul Erdős and Mark Kac on the number of different prime factors of a number $$n\in{\mathbb{N}}$$ n ∈ N . We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Raphaël Salem and Antoni Zygmund.

scholarly journals On the estimation of the rate of convergence in the two-dimensional central limit theorem

1968 ◽  
Vol 8 (3) ◽  
pp. 591-595
Author(s):  
V. Paulauskas ◽  
A. Slušnys
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Two Dimensional

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: В. Паулаускас, А. Слушнис. Оценка скорости сходимости в двумерной центральной предельной теореме V. Paulauskas, A. Slušnys. Konvergavimo greičio įvertinimas dvimatėje centrinėje ribinėje teoremoje

scholarly journals The Random Normal Matrix Model: Insertion of a Point Charge

2021 ◽  
Author(s):  
Yacin Ameur ◽  
Nam-Gyu Kang ◽  
Seong-Mi Seo
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Characteristic Polynomial ◽  
Central Limit ◽  
Point Charge ◽  
Coulomb Gas ◽  
Normal Matrix ◽  
Scaling Limits ◽  
Two Dimensional ◽  
Rotationally Symmetric

AbstractIn this article, we study microscopic properties of a two-dimensional Coulomb gas ensemble near a conical singularity arising from insertion of a point charge in the bulk of the droplet. In the determinantal case, we characterize all rotationally symmetric scaling limits (“Mittag-Leffler fields”) and obtain universality of them when the underlying potential is algebraic. Applications include a central limit theorem for $\log |p_{n}(\zeta )|$ log | p n ( ζ ) | where pn is the characteristic polynomial of an n:th order random normal matrix.

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