The Dedekind sum $s(p,q)$ can be both positive and negative. What are the known lower/upper bounds in terms of p,q? (I would prefer something that grows not faster than q)
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For a fixed $q$, the maximum is $$s(1,q)={1\over4}+{1\over6q}+{q\over12}$$ and the minimum is $s(q1,q)=s(1,q)$. 


See http://www.mathkb.com/Uwe/Forum.aspx/math/38267/upperboundsonDedekindsums (most of the relevant stuff is due to our own @Gerry Myerson) 

