In this paper we consider the problem of exactly evaluating the p-norm of the linear operator linked with arithmetic Dirichlet convolutions. We prove that a simply derived upper bound for this norm is actually attained for several different classes of arithmetic functions including completely multiplicative functions, but not for certain multiplicative functions. Our proof depends fundamentally on the asymptotic distribution properties of smooth numbers.
Smooth numbers and the norms of arithmetic Dirichlet convolutions.
CODECA', Paolo;NAIR, Mohan K.
2008
Abstract
In this paper we consider the problem of exactly evaluating the p-norm of the linear operator linked with arithmetic Dirichlet convolutions. We prove that a simply derived upper bound for this norm is actually attained for several different classes of arithmetic functions including completely multiplicative functions, but not for certain multiplicative functions. Our proof depends fundamentally on the asymptotic distribution properties of smooth numbers.File in questo prodotto:
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