Lutz Lucht

A SURVEY OF RAMANUJAN EXPANSIONS

This paper summarizes the development of Ramanujan expansions of arithmetic functions since Ramanujans paper in 1918, following Carmichaels mean-value-based concept from 1932 up to 1994. A new technique, based on the concept of related arithmetic functions, is introduced that leads to considerable extensions of preceding results on Ramanujan expansions. In particular, very short proofs of theorems for additive and multiplicative functions going far beyond previous borders are presented, and Ramanujan expansions that formerly have been considered mysterious are explained.

An application of Banach algebra techniques to multiplicative functions

I Results and comments The object of this paper is to carry out proofs of weighted versions of Wiener type inversion theorems in the theory of Banach algebras and to derive, as an application, inversion theorems for multiplicative arithmetical functions. Such theorems using general weight functions do not seem to have been known or stated explicitly, until now. For D=2~ or D = N u {0}, denote by W(D) the set of positive valued weight functions (~) on D satisfying (1) cn(0)=l, o(m+n)

Mittelwertungleichungen für Lösungen gewisser Differenzengleichungen

SummaryLet Γ andψ = Γ′/Γ denote the Gamma function and the Psi function respectively. Let furtherλ1,⋯,λn ∈ ℝ+ denote weights,λ1 +⋯+λ = 1. The following pair of inequalities is proved:

Recurrent and almost-periodic sequences

\begin{gathered} \Gamma (x_1^{\lambda _1 } \cdot \cdot \cdot x_n^{\lambda _n } ) \leqslant \Gamma ^{\lambda _1 } (x_1 ) \cdot \cdot \cdot \Gamma ^{\lambda _n } (x_n )(x_1 ,...,x_n \geqslant \alpha ), \hfill \\ \Gamma (x_1^{\lambda _1 } \cdot \cdot \cdot x_n^{\lambda _n } ) \geqslant \Gamma ^{\lambda _1 } (x_1 ) \cdot \cdot \cdot \Gamma ^{\lambda _n } (x_n )(0< x_1 ,...,x_n \leqslant \alpha ) \hfill \\ \end{gathered}

Solutions to arithmetic convolution equations

where α is the unique positive root of the equationψ(α) + αψ′(α) = 0. The first of the above inequalities is also valid for allx1,⋯,xn ∈ ℝ+ under the restraint

Über die Lösungsanzahl der Gleichungf(n)=ϰ·n für gewisse Klassen multiplikativer Funktionen

x_1^{\lambda _1 } \cdot \cdot \cdot x_n^{\lambda _n } \geqslant \beta