ested Logarithm

In number theory (e.g., Ivić 2003), the symbol  is commonly used to mean the nested logarithm (also called the repeated logarithm or iterated logarithm) , where  is the natural logarithm.

Care must be taken based on context as to when the notation  denotes the logarithm to base  and when it means the -nested natural logarithm.

The plots above show , , and  in the complex plane.

The penchant for formulas and bounds containing a profusion of nested logarithms has led to the following joke. What sound does a drowning analytic number theorist make? A: log log log log... (Havil 2003, p. 115).

REFERENCES:

Canadian Mathematical Society. Student Mathematical Communicator. March 2001. http://www.cms.math.ca/Students/en/Newsletter/March2001/cmsletter/node7.html.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.

Ivić, A. "On a Problem of Erdős Involving the Largest Prime Factor of ." 5 Nov 2003. http://arxiv.org/abs/math.NT/0311056.