Mean value theorems for arithmetic functions similar to Euler’s phi-function

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 ...

Arithmetic Functions and the Euler Phi Function

• An arithmetic function takes positive integers as inputs and produces real or complex numbers as outputs. • If f is an arithmetic function, the divisor sum Df(n) is the sum of the values of f at the positive divisors of n. • τ (n) is the number of positive divisors of n; σ(n) is the sum of the positive divisors of n. • The Möbius function μ(n) is 1 if n = 1 and 0 if n has a repeated prime fac...

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

Some Mean Value Theorems for the Riemann Zeta-function and Dirichlet L-functions

The theory of the Riemann zeta-function ζ(s) and Dirichlet L-functions L(s, χ) abounds with unsolved problems. Chronologically the first of these, now known as the Riemann Hypothesis (RH), originated from Riemann’s remark that it is very probable that all non-trivial zeros of ζ(s) lie on the line < s = 12 . Later on Piltz conjectured the same for all of the functions L(s, χ) (GRH). The vertical...