The Distribution of Values of Arithmetic Functions

2013 ◽  
pp. 63-66
Author(s):  
G. V. Fedorov
Keyword(s):  
Arithmetic Functions ◽  
Distribution Of Values

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.

Effective Estimates for the Distribution of Values of Euler Products

2005 ◽  
Vol 145 (2) ◽  
pp. 97-106 ◽  
Author(s):  
Ernesto Girondo ◽  
Jörn Steuding
Keyword(s):  
Distribution Of Values ◽  
Euler Products
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