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Sign changes of error terms related to the Euler function

Part of: Multiplicative number theory

Published online by Cambridge University Press:  26 February 2010

Yuk-Kam Lau
Yuk-Kam Lau
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hone Kong.
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Extract

Let φ(n) be the Euler function (i.e., φ(n) denotes the number of integers less than n which are relatively prime to n), and define

These functions were extensively studied by several mathematicians. One of the problems investigated concerns their sign changes. We say that a function fx) has a sign change at x = x0 if f(x0 −) f(x0 +) < 0, and f(x) has a sign change on the integer n if (n)f(n+1) < 0. The numbers of sign changes and sign changes on integers of f(x) in the interval [1, T] are denoted by Xf(T) and Nf(T), respectively.

MSC classification

Secondary: 11N64: Other results on the distribution of values or the characterization of arithmetic functions
Type
Research Article
Information
Mathematika , Volume 46 , Issue 2 , December 1999 , pp. 391 - 395
Copyright
Copyright © University College London 1999

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References

Ch. Chowla, S., Contributions to the analytic theory of numbers. Math. Z., 35 (1932), 279–299.CrossRefGoogle Scholar
ESI. Erdos, P. and Shapiro, H. M. On the changes of sign of a certain error function. Canad. J. Math. 3 (1951), 375–385.CrossRefGoogle Scholar
ES2. Erdos, P. and Shapiro, H. M. The existence of a distribution function for an error term related to the Euler function. Canad. J. Math., 1 (1955), 63–75.CrossRefGoogle Scholar
Pél. Petermann, Y.-F. S.. Changes of sign of error terms related to Euler's function and to divisor functions. Comm. Math. Helvet., 61 (1986), 84–101.CrossRefGoogle Scholar
Pé2. Petermann, Y.-F. S.. Changes of sign of error terms related to Euler's function and to divisor functions II. Ada Arith., 51 (1988), 321–333.CrossRefGoogle Scholar
Pe3. Pétermann, Y.-F. S.. On the distribution of values of an error term related to the Euler function. Théorie des nombres (Quebec, PQ, 1987), (de Gruyter, Berlin, 1989), pp. 785–797.Google Scholar