Mean Value theorems for multiplicative functions bounded in mean α-power, α >1

Mean-value Theorems for Multiplicative Arithmetic Functions of Several Variables

Let f : Nn → C be an arithmetic function of n variables, where n ≥ 2. We study the mean-value M(f) of f that is defined to be lim x1,...,xn→∞ 1 x1 · · ·xn ∑ m1≤x1, ... , mn≤xn f(m1, . . . , mn), if this limit exists. We first generalize the Wintner theorem and then consider the multiplicative case by expressing the mean-value as an infinite product over all prime numbers. In addition, we study ...

Harmonic functions via restricted mean-value theorems

Let f be a function on a bounded domain Ω ⊆ R and δ be a positive function on Ω such that B(x, δ(x)) ⊆ Ω. Let σ(f)(x) be the average of f over the ball B(x, δ(x)). The restricted mean-value theorems discuss the conditions on f, δ, and Ω under which σ(f) = f implies that f is harmonic. In this paper, we study the stability of harmonic functions with respect to the map σ. One expects that, in gen...

Liftings and Mean Value Theorems for Automorphic L-functions

Mean values of multiplicative functions

Let f(n) be a totally multiplicative function such that |f(n)| ≤ 1 for all n, and let F (s) = ∑∞ n=1 f(n)n−s be the associated Dirichlet series. A variant of Halász’s method is developed, by means of which estimates for ∑N n=1 f(n)/n are obtained in terms of the size of |F (s)| for s near 1 with 1. The result obtained has a number of consequences, particularly concerning the zeros of the p...

Inequalities for J−contractions Involving the Α−power Mean

A selfadjoint involutive matrix J endows C with an indefinite inner product [·, ·] given by [x, y] := 〈Jx, y〉, x, y ∈ C. We present some inequalities of indefinite type involving the α−power mean and the chaotic order. These results are in the vein of those obtained by E. Kamei [6, 7].