A number n is called a barrier of a number-theoretic function f(m) if, for all m<n, m+f(m)<=n. Neither the totient function phi(n) nor the divisor function sigma(n) has a barrier.

Let U subset= C be an open set and x_0 in partialU, then a function b:U^_->R is called a barrier for U at a point x_0 if

1. b is continuous,

2. b is subharmonic on U,

3. b|_(partialU)<=0,

4. {z in partialU:b(z)=0}={z_0}

(Krantz 1999, pp. 100-101).

See also

Subharmonic Function

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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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Cite this as:

Weisstein, Eric W. "Barrier." From MathWorld--A Wolfram Web Resource.

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