Nested 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).