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.
