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James Huard - An arithmetic reciprocity relation of Liouville type and applications



JAMES HUARD, Department of Mathematics, Canisius College, Buffalo, New York  14208, USA
An arithmetic reciprocity relation of Liouville type and applications


Let n be a positive integer and let $f\colon {\bf Z}\rightarrow {\bf
C}$ be an even function. Liouville gave the identity
\begin{align*}\sum_{aA + bB = n} \bigl( f(a-b) -f (a+b)\bigr)
&\null= f(0) \bigl...
...sum_{d\vert n} df(d) - 2\sum_{d\vert n} \sum_{1\leq v \leq d}
f(v),
\end{align*}
where the sum on the left hand side is taken over all positive integers
a, A, b, B satisfying aA + bB =n. Some generalizations of this identity will be presented and applications will be made to the determination of such sums as

\begin{displaymath}\sum^{n-1}_{m=1} \sigma_3 (m)\sigma_3 (n-m)
\end{displaymath}

and to the representation of n by certain positive quaternary quadratic forms. The proofs of these results require elementary methods. This is a joint work with Z. Ou, B. K. Spearman, and K. S. Williams.


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