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Ω-estimates for a class of arithmetic error terms

Published online by Cambridge University Press:  01 May 2007

JERZY KACZOROWSKI  and
KAZIMIERZ WIERTELAK
JERZY KACZOROWSKI
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland.
KAZIMIERZ WIERTELAK
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland.
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Abstract

The main aim of this paper is to present a general method of proving Ω-estimates for a class of arithmetic error terms. We assume that error terms in question are boundary values of harmonic functions on the upper half-plane satisfying certain subsidiary conditions. We prove a general theorem for an axiomatically defined class of such functions and then we show how this result can be used to give statements in concrete situations. As examples we treat the classical case of the remainder term in the prime number formula obtaining a new proof of the well-known result of J. E. Littlewood, and the case of the remainder term in the asymptotic formula for the summatory function of the square-free divisor function. In the latter case our result is new.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 3 , May 2007 , pp. 385 - 394
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

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