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March 13th, 2011, 06:37 PM   #1
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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?!
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March 13th, 2011, 07:52 PM   #2
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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.
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March 14th, 2011, 06:16 PM   #3
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Re: Function between constant functions and iterated logarit

Thanks a lot!!! It's really a nice function which possesses the properties I need!
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March 14th, 2011, 06:31 PM   #4
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Re: Function between constant functions and iterated logarit

Glad to help! Out of curiosity, what is this for?
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March 16th, 2011, 07:05 PM   #5
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Re: Function between constant functions and iterated logarit

I used this in some work about the riemann zeta function
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