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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 k-th 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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