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Ω-estimates for a class of arithmetic error terms

Published online by Cambridge University Press:  01 May 2007

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.

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
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Baker, R. C.. The square-free divisor problem. Quart. J. Math.(Oxford) (2) 45 (1994), 269277.CrossRefGoogle Scholar
[2]Baker, R. C.. The square-free divisor problem II. Quart. J. Math. (Oxford) (2) 47 (1996), 133146.Google Scholar
[3]Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.. Higher Transcendental Functions, Vol II (McGraw-Hill, 1953).Google Scholar
[4]Kaczorowski, J.. The k-functions in multiplicative number theory, I; On complex explicit formulae. Acta Arith. 56 (1990), 195211.Google Scholar
[5]Kaczorowski, J.. The k-functions in multiplicative number theory, IV; On a method of A.E. Ingham. Acta Arith. 57 (1991), 231244.CrossRefGoogle Scholar
[6]Kaczorowski, J.. Boundary values of Dirichlet series and the distribution of primes. Progr. Math. 168 (1998), 237254.CrossRefGoogle Scholar
[7]Littlewood, J. E.. Sur la distribution des nombres premiers. C. R. Acad. Sci. Paris. 158 (1914), 18691872.Google Scholar
[8]Nowak, W. G. and Schmeier, M.. Conditional asymptotic formulae for a class of arithmetic functions. Proc. Amer. Math. Soc. 103 (1988), 713717.CrossRefGoogle Scholar
[9]Saffari, B.. Sur le nombre de diviseurs r-libres d'un entier, et sur les points à coordonnées entières dans certaines régions du plan. C. R. Acad. Sci. Paris Sér. A-B. 266 (1968), A601A603.Google Scholar
[10]Suryanarayana, D. and Prasad, V. Siva Rama. The number of k-free divisors of an integer. Acta Arith. 17 (1970/71), 345354.Google Scholar
[11]Wiertelak, K.. On some connections between zeta-zeros and square-free divisors of an integer. Functiones et Approximatio. 31 (2003), 133145.Google Scholar
[12]Wu, Jie. On the primitive circle problem. Monatsh. Math. 135 (2002), 6981.CrossRefGoogle Scholar