On some generalizations of mean value theorems for arithmetic functions\n of two variables

2016/04/18 by Noboru Ushiroya, Ushiroya, Noboru
Mathematics · #11N37 #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1604.05410

openalex publication_date 2016/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f: \ℕ² \↦ \ℂ be an arithmetic function of two\nvariables. We study the existence of the limit:\n \
displaystyle
lim_x
to
infty
frac1x² (
log x)^k-1
sum_{n1 ,\nn2
le x} f (n₁, n₂) where k is a fixed positive integer. Moreover, we\nexpress this limit as an infinite product over all prime numbers in the case\nthat f is a multiplicative function of two variables. This study is a\ngeneralization of Cohen-van der Corput's results to the case of two variables.\n

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