The image of the generalized Dedekind sum
Evelyne S. Knight, Carlos Alexov Matos, Amira Sefidi, Matthew P. Young
TL;DR
This work analyzes the image of the newform Dedekind sum $S_{\chi_1,\chi_2}$ on $\Gamma_1(q_1q_2)$, showing it forms a lattice in $F_{\chi_1,\chi_2}$ and establishing explicit lattice containment bounds such as $S_{\chi_1,\chi_2}(\Gamma_1(q_1q_2)) \subseteq \frac{1}{\gcd(q_1,q_2)}\mathbb{Z}[\chi_1,\chi_2]$ and, for quadratic $\chi_1,\chi_2$ with odd $q_1,q_2>4$, $S_{\chi_1,\chi_2}(\Gamma_0(q_1q_2)) \subseteq \frac{1}{\gcd(q_1,q_2)}\mathbb{Z}$. The paper gives an explicit denominator bound showing $S_{\chi_1,\chi_2}(a,c)$ has denominator $\frac{1}{rq_1}$ when $c=rq_1q_2$, and derives an explicit formula in terms of floor sums. Using a reciprocity law and number-theoretic tools (e.g., class number relations for odd quadratic characters), the authors prove containment results that in certain quadratic-odd regimes tighten to $\frac{1}{\gcd(q_1,q_2)}\mathbb{Z}$ or even $\mathbb{Z}$ when conductors are coprime and large. These results advance the conjecture that for quadratic characters the image is a small lattice, and motivate the proposed generalized two-conjecture. The work contributes concrete denominator control, reciprocity-based refinements, and data-driven guidance toward a complete description of the image lattice.
Abstract
The newform Dedekind sum $S_{χ_1, χ_2}$ associated to a pair of primitive Dirichlet characters $χ_1$, $χ_2$ of respective conductors $q_1$, $q_2$, is a group homomorphism from $Γ_1(q_1 q_2)$ into the number field $F_{χ_1, χ_2}$ generated by the values of the characters. It is a basic question to identify the image of this map, which is known to be a lattice $L_{χ_1, χ_2}$ in $F_{χ_1, χ_2}$. It has recently been conjectured that when $χ_1$ and $χ_2$ are quadratic, then $ L_{χ_1, χ_2} = 2 \mathbb{Z}$. In this paper, we make some progress towards this conjecture by exhibiting an explicit lattice in which $L_{χ_1, χ_2}$ is contained; in particular, when the characters are quadratic, the $q_i$ are coprime, odd, and sufficiently large, then $L_{χ_1, χ_2} \subseteq \mathbb{Z}$.