Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2512.21413 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912788966801408 |
|---|---|
| author | Fedosova, Ksenia Klinger-Logan, Kim |
| author_facet | Fedosova, Ksenia Klinger-Logan, Kim |
| contents | We find exact identities for sums of the form \begin{equation*}\label{eq:convsumabs}
\sum_{\stackrel{n_1+n_2 = n}{n_1 \in \mathbb{Z} \setminus \{ 0, n \} }} Q(n_1,n_2) σ_{-r_1}(n_1) σ_{-r_2}(n_2),
\end{equation*}
where $n\in\mathbb{N}$, $r_1,r_2\in\mathbb{C}$, $Q$ is a combination of hypergeometric functions, and $σ_{a}(x)$ denotes the divisor function. Specifically,
we find that they can be expressed in terms of Fourier coefficients of Hecke cusp forms weighted by their $L$-values. This result expands upon previous work with Radchenko in which such identities were found for divisor functions with even integer index \cite{FKLR} and encompasses results of Jacobi \cite{motohashi1994binary} and Diamantis and O'Sullivan in \cite{diamantis2010kernels, o2023identities} for divisor functions with odd integer index. The proof of our result expresses these sums in terms of Estermann zeta functions and uses trace formulae. In addition, we use a regularization of divergent convolution sums to provide a mathematical explanation for $L$-values (non-critical in the sense of Deligne) appearing in modular graph functions \cite{DKS2021_2}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21413 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convolution identities for complex-indexed divisor functions and modular graph functions Fedosova, Ksenia Klinger-Logan, Kim Number Theory Mathematical Physics We find exact identities for sums of the form \begin{equation*}\label{eq:convsumabs} \sum_{\stackrel{n_1+n_2 = n}{n_1 \in \mathbb{Z} \setminus \{ 0, n \} }} Q(n_1,n_2) σ_{-r_1}(n_1) σ_{-r_2}(n_2), \end{equation*} where $n\in\mathbb{N}$, $r_1,r_2\in\mathbb{C}$, $Q$ is a combination of hypergeometric functions, and $σ_{a}(x)$ denotes the divisor function. Specifically, we find that they can be expressed in terms of Fourier coefficients of Hecke cusp forms weighted by their $L$-values. This result expands upon previous work with Radchenko in which such identities were found for divisor functions with even integer index \cite{FKLR} and encompasses results of Jacobi \cite{motohashi1994binary} and Diamantis and O'Sullivan in \cite{diamantis2010kernels, o2023identities} for divisor functions with odd integer index. The proof of our result expresses these sums in terms of Estermann zeta functions and uses trace formulae. In addition, we use a regularization of divergent convolution sums to provide a mathematical explanation for $L$-values (non-critical in the sense of Deligne) appearing in modular graph functions \cite{DKS2021_2}. |
| title | Convolution identities for complex-indexed divisor functions and modular graph functions |
| topic | Number Theory Mathematical Physics |
| url | https://arxiv.org/abs/2512.21413 |