An Arithmetic Sum Associated with the Classical Theta Function

The sum $S(h,k):={\sum }_{j=1}^{k-1}(-1{)}^{j+1+[hj\mathrm{/}k]}$ appears in the modular transformation formulae of the classical theta function ${\vartheta }_3(z)$. The double sum $S(k):={\sum }_{h=1}^{k-1}S(h,k)$ has a remarkable distribution of values. Although properties for $S(k)$ and a related sum can be established, several interesting conjectures are open.