Uniform Distribution of Values of Multiplicative Functions

Wenbin Zhang
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Wenbin Zhang: South China University of Technology Guangzhou, Department of Mathematics

A chapter in International Symposium in Memory of Hua Loo Keng, 1991, pp 347-353 from Springer

Abstract: Abstract Let n → ϕ(n) be a positive valued arithmetic function which tends to infinity as n → ∞. Following [5], we shall say that the values of ϕ are uniformly distributed in (0, ∞) (briefly, ϕ is u. d. in (0, ∞)) if there exists a positive constant c such that % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaabm % aabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaad6eadaqadaqaaiaa % dIhacaGG7aGaeqOXdOgacaGLOaGaayzkaaGaaiOoaiabg2da9iaaco % cadaGadaqaaiaad6gacqGHflY1cqaHgpGAdaqadaqaaiaad6gaaiaa % wIcacaGLPaaacqGHKjYOcaWG4baacaGL7bGaayzFaaGaeSipIOJaam % 4yaiaadIhaaaa!51FB! $$N\left( x \right) = N\left( {x;\varphi } \right): = \# \left\{ {n \cdot \varphi \left( n \right) \leqslant x} \right\} \sim cx$$ as x → ∞. The number c will be called the density of values of ϕ. Also, we shall say that the values of ϕ are distributed with zero (respectively infinite) density in (0, ∞) if N(x)/x tends to zero (respectively infinity) as x → ∞.

Keywords: Multiplicative Function; Arithmetic Function; Zero Density; Generalize Integer; Integral Version (search for similar items in EconPapers)
Date: 1991
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DOI: 10.1007/978-3-662-07981-2_21

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