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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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2 Answers

up vote 6 down vote accepted

For a fixed $q$, the maximum is $$s(1,q)=-{1\over4}+{1\over6q}+{q\over12}$$ and the minimum is $s(q-1,q)=-s(1,q)$.

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Wow, that's great. And 1/12 is especially delightful. Thanks! Can you also give a reference, what to cite? –  Dmitry Kerner May 4 '11 at 12:55
    
This follows from a finite-Fourier-series version of Cauchy-Schwartz. The earliest reference I'm aware of is H. Rademacher, Zur Theorie der Dedekindschen Summen, Math. Z. 63 (1956) 445-463. 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>). –  matthias beck May 4 '11 at 15:26
    
$s(p,q)=q^{-2}\sum af(a)$ plus terms not relevant here, where $f(a)$ is the permutation of $1,2,\dots,q-1$ 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 $1,2,\dots,q-1$ gives $q-1,\dots,2,1$, and this corresponds to multiplication by $q-1$. –  Gerry Myerson May 5 '11 at 0:20
    
Thanks a lot for both comments! –  Dmitry Kerner May 5 '11 at 11:26
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See

http://www.mathkb.com/Uwe/Forum.aspx/math/38267/upper-bounds-on-Dedekind-sums

(most of the relevant stuff is due to our own @Gerry Myerson)

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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... –  Dmitry Kerner May 4 '11 at 0:04
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