An Application of Free Probability to Arithmetic Functions

2014 ◽  
Vol 9 (7) ◽  
pp. 1457-1489 ◽  
Author(s):  
Ilwoo Cho ◽  
Palle E. T. Jorgensen
Keyword(s):  
Free Probability ◽  
Arithmetic Functions

Correction: An Application of Free Probability to Arithmetic Functions

2016 ◽  
Vol 12 (7) ◽  
pp. 1567-1608
Author(s):  
Ilwoo Cho ◽  
Palle E. T. Jorgensen
Keyword(s):  
Free Probability ◽  
Arithmetic Functions

Asymptotic free probability for arithmetic functions and factorization of Dirichlet series

2015 ◽  
Vol 6 (3) ◽  
pp. 255-295 ◽  
Author(s):  
Ilwoo Cho ◽  
Timothy Gillespie ◽  
Palle E. T. Jorgensen
Keyword(s):  
Dirichlet Series ◽  
Free Probability ◽  
Arithmetic Functions

scholarly journals On the growth and zeros of polynomials attached to arithmetic functions

2021 ◽  
Author(s):  
Bernhard Heim ◽  
Markus Neuhauser
Keyword(s):  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Fourier Coefficients ◽  
Arithmetic Functions ◽  
Zeros Of Polynomials ◽  
Simple Complex ◽  
Hook Length ◽  
Moderate Growth ◽  
Growth Properties ◽  
Sum Of Divisors

AbstractIn this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and $$0<h(n) \le h(n+1)$$ 0 < h ( n ) ≤ h ( n + 1 ) . We put $$P_0^{g,h}(x)=1$$ P 0 g , h ( x ) = 1 and $$\begin{aligned} P_n^{g,h}(x) := \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{aligned}$$ P n g , h ( x ) : = x h ( n ) ∑ k = 1 n g ( k ) P n - k g , h ( x ) . As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind $$\eta $$ η -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind.

Limit periodic arithmetic functions

1985 ◽  
Vol 25 (3) ◽  
pp. 243-250 ◽  
Author(s):  
Z. Kryžius
Keyword(s):  
Arithmetic Functions
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