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Date: 3-7-2018
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A number is called a barrier of a number-theoretic function if, for all , . Neither the totient function nor the divisor function has a barrier.
Let be an open set and , then a function is called a barrier for at a point if
1. is continuous,
2. is subharmonic on ,
3. ,
4.
(Krantz 1999, pp. 100-101).
REFERENCES:
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 64-65, 1994.
Krantz, S. G. "The Concept of a Barrier." §7.7.9 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 100-101, 1999.
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