Some asymptotic formulae involving powers of arithmetic functions

1989 ◽  
pp. 201-234 ◽  
Author(s):  
V. Sitaramaiah ◽  
M. V. Subbarao
Keyword(s):  
Arithmetic Functions ◽  
Asymptotic Formulae

scholarly journals Counting in hyperbolic spikes: The diophantine analysis of multihomogeneous diagonal equations

2018 ◽  
Vol 2018 (737) ◽  
pp. 255-300
Author(s):  
Valentin Blomer ◽  
Jörg Brüdern
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Arithmetic Functions ◽  
New Method ◽  
Algebraic Varieties ◽  
Asymptotic Formulae ◽  
Versatile Tool

AbstractA method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a versatile tool to count rational points on algebraic varieties defined by multi-homogeneous diagonal equations.

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

2021 ◽  
pp. 1-16
Author(s):  
Alexander Dabrowski
Keyword(s):  
General Type ◽  
Differential Operators ◽  
Laplacian Operator ◽  
Variational Characterization ◽  
Repeated Eigenvalues ◽  
Direct Extension ◽  
Asymptotic Formulae ◽  
Elliptic Differential Operators ◽  
Domain Perturbations ◽  
Boundary Deformation

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

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

On asymptotic formulae via summability

2011 ◽  
Vol 81 (10) ◽  
pp. 2174-2180 ◽  
Author(s):  
P. Garrancho ◽  
D. Cárdenas-Morales ◽  
F. Aguilera
Keyword(s):  
Asymptotic Formulae
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