On the usual product of rational arithmetic functions

Rational arithmetic functions are arithmetic functions of the formg1∗⋯∗gr∗h1−1∗⋯∗hs−1, wheregi,hjare completely multiplicative functions and∗denotes the Dirichlet convolution. Four aspects of these functions are studied. First, some characterizations of such functions are established; second, possible Busche-Ramanujan-type identities are investigated; third, binomial-type identities are derived; and finally, properties of the Kesava Menon norm of such functions are proved.

Download Full-text

The Multi-Dimensional Local Theorem for Additive Arithmetic Functions

Analytic and Probabilistic Methods in Number Theory ◽

10.1515/9783112314234-029 ◽

1992 ◽

pp. 285-292

Keyword(s):

Arithmetic Functions ◽

Additive Arithmetic ◽

Local Theorem

Download Full-text

On the growth and zeros of polynomials attached to arithmetic functions

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg ◽

10.1007/s12188-021-00241-3 ◽

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.

Download Full-text

Semi-invariant Riemannian submersions from nearly Kaehler manifolds

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887820501005 ◽

2020 ◽

Vol 17 (07) ◽

pp. 2050100

Author(s):

Rupali Kaushal ◽

Rashmi Sachdeva ◽

Rakesh Kumar ◽

Rakesh Kumar Nagaich

Keyword(s):

Sufficient Conditions ◽

Riemannian Submersion ◽

Horizontal Distribution ◽

Product Manifold ◽

Riemannian Submersions ◽

Kaehler Manifolds ◽

Necessary And Sufficient ◽

Nearly Kaehler Manifold ◽

The Vertical Distribution ◽

Usual Product

We study semi-invariant Riemannian submersions from a nearly Kaehler manifold to a Riemannian manifold. It is well known that the vertical distribution of a Riemannian submersion is always integrable therefore, we derive condition for the integrability of horizontal distribution of a semi-invariant Riemannian submersion and also investigate the geometry of the foliations. We discuss the existence and nonexistence of semi-invariant submersions such that the total manifold is a usual product manifold or a twisted product manifold. We establish necessary and sufficient conditions for a semi-invariant submersion to be a totally geodesic map. Finally, we study semi-invariant submersions with totally umbilical fibers.

Download Full-text