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Mean values of arithmetic functions and application to sums of powers

Published online by Cambridge University Press:  08 August 2025

RÉGIS DE LA BRETÈCHE
Affiliation:
Université Paris Cité, Sorbonne Université, CNRS, Institut Universitaire de France, Institut de mathématiques de Jussieu-Paris Rive Gauche, F-75013 Paris, France. e-mail: regis.delabreteche@imj-prg.fr
GÉRALD TENENBAUM
Affiliation:
Institut Élie Cartan Université de Lorraine, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France. e-mail: gerald.tenenbaum@univ-lorraine.fr

Abstract

We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erdős—Hooley $\Delta$-function, we derive lower bounds for the cardinality of those integers not exceeding a given limit that are expressible as certain sums of powers.

Information

Type
Research Article
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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