Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables

Integers ◽  
2012 ◽  
Vol 12 (5) ◽  
Author(s):  
Noboru Ushiroya
Keyword(s):  
Infinite Product ◽  
Prime Numbers ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Mean Value Theorems ◽  
Several Variables ◽  
The Mean ◽  
Multiplicative Arithmetic

Abstract.Letif 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 the mean-value of a function of the form

Multiplicative functions at consecutive integers. II

1988 ◽  
Vol 103 (3) ◽  
pp. 389-398 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Sufficient Conditions ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Multiplicative Functions ◽  
Global Behaviour ◽  
Consecutive Integers ◽  
The Mean ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Multiplicative Arithmetic

The global behaviour of multiplicative arithmetic functions has been extensively studied and is now well understood for a large class of multiplicative functions. In particular, Halász [5] completely determined the asymptotic behaviour of the meansfor multiplicative functions g satisfying |g| ≤ 1, and gave necessary and sufficient conditions for the existence of the ‘mean value’

scholarly journals Mean value theorems for a class of density-like arithmetic functions

2020 ◽  
pp. 1-15
Author(s):  
Lucas Reis
Keyword(s):  
Finite Field ◽  
Arithmetic Function ◽  
Field Extension ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Mean Value Theorem ◽  
Primitive Elements ◽  
Mean Values ◽  
Mean Value Theorems ◽  
Nonzero Mean

This paper provides a mean value theorem for arithmetic functions [Formula: see text] defined by [Formula: see text] where [Formula: see text] is an arithmetic function taking values in [Formula: see text] and satisfying some generic conditions. As an application of our main result, we prove that the density [Formula: see text] (respectively, [Formula: see text]) of normal (respectively, primitive) elements in the finite field extension [Formula: see text] of [Formula: see text] are arithmetic functions of (nonzero) mean values.

Some weighted quadrature methods based upon the mean value theorems

2020 ◽  
Author(s):  
Herbert H. H. Homeier ◽  
Hari M. Srivastava ◽  
Mohammad Masjed‐Jamei ◽  
Zahra Moalemi
Keyword(s):  
Mean Value ◽  
Mean Value Theorems ◽  
Quadrature Methods ◽  
The Mean ◽  
Weighted Quadrature

Mean value theorems for polynomial solutions of linear elliptic equations with constant coefficients in the complex plane

2020 ◽  
Vol 17 (4) ◽  
pp. 594-600
Author(s):  
Olga Trofymenko
Keyword(s):  
Elliptic Equation ◽  
Complex Plane ◽  
Elliptic Equations ◽  
Regular Polygon ◽  
Mean Value ◽  
Linear Elliptic Equation ◽  
Mean Value Theorems ◽  
Polynomial Solutions ◽  
Constant Coefficients ◽  
The Mean

We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon.

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