• Corpus ID: 119262395

Random matrix-valued multiplicative functions and linear recurrences in Hilbert-Schmidt norms of random matrices

@article{Gerspach2018RandomMM,
  title={Random matrix-valued multiplicative functions and linear recurrences in Hilbert-Schmidt norms of random matrices},
  author={Maxim Gerspach},
  journal={arXiv: Number Theory},
  year={2018}
}
  • Maxim Gerspach
  • Published 11 December 2018
  • Mathematics, Computer Science
  • arXiv: Number Theory
We introduce the notion of a random matrix-valued multiplicative function, generalizing Rademacher random multiplicative functions to matrices. We provide an asymptotic for the second moment based on a linear recurrence property for Hilbert-Schmidt norms of sucessive products of random matrices. Moreover, we provide upper bounds for the higher even moments related to the generalized joint spectral radius. 

References

SHOWING 1-10 OF 11 REFERENCES

Moments of random multiplicative functions and truncated characteristic polynomials

We give an asymptotic formula for the $2k$th moment of a sum of multiplicative Steinhaus variables. This was recently computed independently by Harper, Nikeghbali and Radziwi\l\l. We also compute the

A note on Helson's conjecture on moments of random multiplicative functions

We give lower bounds for the small moments of the sum of a random multiplicative function, which improve on some results of Bondarenko and Seip and constitute further progress towards (dis)proving a

Computationally Efficient Approximations of the Joint Spectral Radius

TLDR
A procedure for approximating the joint spectral radius of a finite set of matrices with arbitrary high accuracy based on semidefinite liftings and it is proved that a widely used approximation based on common quadratic Lyapunov functions (or on ellipsoid norms) has relative accuracy $1/\sqrt m$, where $m$ is the number ofMatrices and $n$ is their size.

On a generalized divisor problem I

  • Y. Lau
  • Mathematics
    Nagoya Mathematical Journal
  • 2002
We give a discussion on the properties of Δ a (x) (− 1 < a < 0), which is a generalization of the error term Δ(x) in the Dirichlet divisor problem. In particular, we study its oscillatory nature and

On the joint spectral radius

G. Panti, Pythagorean triples, billiards, and mean free paths. We show that primitive pythagorean triples can be enumerated by playing billiard on a mgonal billiard table in the Poincaré disk. The

Generalized divisor problem

In 1952 H.E. Richert by means of the theory of Exponents Pairs (developed by J.G. van der Korput and E. Phillips ) improved the above O-term ( see [8] or [4] pag. 221 ). In 1969 E. Kratzel studied

Estimation de sommes multiples de fonctions arithmétiques

We estimate some sums of the shape S(Xβ1,…, Xβm): = [sum ] 1 [les ] d1 [les ] Xβ1… [sum ]1 [les ] dm [les ] Xβm ƒ(d1,…, dm), when m ∈ $\Bbb N$ and f is a nonnegative arithmetical function. We relate

Introduction To Analytic And Probabilistic Number Theory

TLDR
This introduction to analytic and probabilistic number theory is free to download and may help people find their favorite readings that end up in malicious downloads.