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March 13th, 2011, 05:37 PM  #1 
Newbie Joined: Mar 2011 Posts: 10 Thanks: 0  Function between constant functions and iterated logarithms
I am looking for a continuous and increasing function f(x) which tends to infinity as x tends to infinity. This function must have the property that it is eventually smaller than log_k(x) (the kth iterated logarithm) for all k>=1 I have no hint how to find such a function! One of my problems is that log_k(x) tends to 0 when k tends to infinity... then how is it possible to find f(x) increasing?! 
March 13th, 2011, 06:52 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Function between constant functions and iterated logarit
There's a function called log* that counts the number of times the logarithm must be taken to make the argument <= 1. This grows more slowly than any fixed number of logarithms, e.g. slower than log(log(log x)). Edit: You can make this continuous by adding the result after the logs. So the function would be x for x <= 1, 1 + log x for 1 < x <= e, 2 + log log x for e < x <= e^e, etc. Lots of other possibilities exist, of course. 
March 14th, 2011, 05:16 PM  #3 
Newbie Joined: Mar 2011 Posts: 10 Thanks: 0  Re: Function between constant functions and iterated logarit
Thanks a lot!!! It's really a nice function which possesses the properties I need!

March 14th, 2011, 05:31 PM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Function between constant functions and iterated logarit
Glad to help! Out of curiosity, what is this for?

March 16th, 2011, 06:05 PM  #5 
Newbie Joined: Mar 2011 Posts: 10 Thanks: 0  Re: Function between constant functions and iterated logarit
I used this in some work about the riemann zeta function


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