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 $s(q1,q)=s(1,q)$.

$\begingroup$ Wow, that's great. And 1/12 is especially delightful. Thanks! Can you also give a reference, what to cite? $\endgroup$ – Dmitry Kerner May 4 '11 at 12:55

$\begingroup$ This follows from a finiteFourierseries version of CauchySchwartz. The earliest reference I'm aware of is H. Rademacher, Zur Theorie der Dedekindschen Summen, Math. Z. 63 (1956) 445463. With some work, one can obtain even better bounds (see, e.g., <a href="front.math.ucdavis.edu/math.NT/0305421">my paper with S. Robins and S. Zacks</a>). $\endgroup$ – matthias beck May 4 '11 at 15:26

$\begingroup$ $s(p,q)=q^{2}\sum af(a)$ plus terms not relevant here, where $f(a)$ is the permutation of $1,2,\dots,q1$ induced by multiplication by $p$ and reduction mod $q$. The maximum over all permutations $f$ (not just those arising from multiplication) of $\sum af(a)$ is attained when $f(a)=a$ for all $a$, and this corresponds to multiplication by 1; the minimum, when $f$ applied to $1,2,\dots,q1$ gives $q1,\dots,2,1$, and this corresponds to multiplication by $q1$. $\endgroup$ – Gerry Myerson May 5 '11 at 0:20

$\begingroup$ Is there a known bound in terms of $s(1,q)$? I'm interested in a bound of the type $s(a,q) < c\cdot s(1,q)$, for some constant $c$ and any $1<a<q1$. $\endgroup$ – jiyanez Jul 5 '16 at 2:42

$\begingroup$ @jiyanez, $c=1$ will do, of course. I think for fixed $q$ large enough and $1<a<q1$ the maximum value of $s(a,q)$ is $s(2,q)$, which is not hard to evaluate. $\endgroup$ – Gerry Myerson Jul 5 '16 at 6:15
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See
http://www.mathkb.com/Uwe/Forum.aspx/math/38267/upperboundsonDedekindsums
(most of the relevant stuff is due to our own @Gerry Myerson)

$\begingroup$ Sorry, I don't see any bound there, only some discussion on properties of the function. I'm completely ignorant in NT, I didn't mean some recent state of art result. There certainly must be some classical rough bound... $\endgroup$ – Dmitry Kerner May 4 '11 at 0:04

$\begingroup$ The link seems to be kaput. kb, that's Kevin Brown, now uses mathpages.com/home but I couldn't find the Dedekind sums essay there. $\endgroup$ – Gerry Myerson Jul 5 '16 at 22:48

$\begingroup$ I found a sci.math thread titled "upper bounds on Dedekind sums" at groups.google.com/forum/#!searchin/sci.math/… and maybe the kb essay is drawn from it. $\endgroup$ – Gerry Myerson Jul 5 '16 at 23:00